CFD Modeling and Analysis of a Planar Anode
Supported Intermediate Temperature
Solid Oxide Fuel Cell
Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF SCIENCE
Major Subject: Mechanical Engineering
by
Melissa Tweedie
May, 2014
Rensselaer Polytechnic Institute
Hartford, Connecticut
Copyright 2014
By
Melissa Tweedie
All Rights Reserved
ii
Contents
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
ABSTRACT ................................................................................................................... viii
NOMENCLATURE ......................................................................................................... ix
1
Introduction.................................................................................................................. 1
2
Methodology ................................................................................................................ 6
3
4
5
2.1
Domain and Physical Parameters ....................................................................... 6
2.2
Operating Conditions ......................................................................................... 8
2.3
CFD Model Overview ........................................................................................ 9
Momentum Model ....................................................................................................... 9
3.1
General Equations ............................................................................................ 10
3.2
Density and Viscosity ...................................................................................... 10
3.3
Microstructural Properties................................................................................ 12
Mass Transfer Model ................................................................................................. 14
4.1
General Equations ............................................................................................ 14
4.2
Maxwell-Stefan Diffusivity ............................................................................. 15
Heat Transfer Model .................................................................................................. 16
5.1
General Equations ............................................................................................ 16
5.1.1. Flow Fields ........................................................................................... 16
5.1.2. Electrodes, Electrolyte and Interconnects ........................................... 19
5.2
Heat Generation Source Terms ........................................................................ 20
5.2.1. Heat Generated by Reactions ............................................................... 20
5.2.2. Heat Generation from Polarizations ..................................................... 21
6
Chemical Model......................................................................................................... 22
6.1
Internal Reforming ........................................................................................... 22
6.2
Chemical Species Balance Equations .............................................................. 23
ii
7
6.3
WGS Kinetics .................................................................................................. 23
6.4
Carbon Deposition ........................................................................................... 24
6.5
Additional Chemical Model Information ......................................................... 25
Electrochemical Model .............................................................................................. 25
7.1
Approaches to Electrochemical Modeling ....................................................... 25
7.2
Electrochemical Species Balance Equations .................................................... 26
7.3
Ion and Charge Transfer................................................................................... 27
7.3.1. Electrode Backing Layers .................................................................... 28
7.3.2. Electrochemical Reaction Layers (ERL) ............................................. 28
7.3.3. Electrolyte ............................................................................................ 29
8
9
7.4
Cell Voltage ..................................................................................................... 30
7.5
Activation Polarizations ................................................................................... 31
7.6
Ohmic Polarizations ......................................................................................... 34
7.7
Concentration Polarizations ............................................................................. 35
Results........................................................................................................................ 36
8.1
Solution Method ............................................................................................... 36
8.2
Comparison to Previous Studies ...................................................................... 37
8.3
Velocity and Pressure Profiles ......................................................................... 39
8.4
Species Distribution ......................................................................................... 41
8.5
Temperature ..................................................................................................... 54
8.6
Electrochemistry .............................................................................................. 55
Conclusion ................................................................................................................. 63
10 Future Work ............................................................................................................... 64
11 References.................................................................................................................. 66
12 Appendix A................................................................................................................ 71
13 Appendix B ................................................................................................................ 76
14 Appendix C ................................................................................................................ 78
iii
LIST OF TABLES
Table 1 Types of Fuel Cells .............................................................................................. 1
Table 2 Cell Dimensions .................................................................................................. 7
Table 3 Cell Materials ....................................................................................................... 7
Table 4 Base Case Physical Properties and Parameters .................................................... 8
Table 5 Model Operating Conditions ................................................................................ 8
Table 6 Simulated Fuel Feed Mole Fractions .................................................................... 9
Table 7 Species Dynamic Viscosity Coefficients ........................................................... 12
Table 8 Ni-YSZ Anode Microstructural Characteristics in Literature ............................ 13
Table 9 Fuller Diffusion Volume ................................................................................... 15
Table 10 Species Heat Capacity Coefficients ................................................................. 17
Table 11 Species Thermal Conductivity Coefficients .................................................... 19
Table 12 Types of SOFC Heat Sources .......................................................................... 20
Table 13 Summary of Heat Source Equations used in Model ......................................... 21
Table 14 Summary of Chemical Species Balance Equations used in Model .................. 23
Table 15 Summary of Electrochemical Species Balance Equations used in Model ....... 27
Table 16 Summary of Charge Transfer Equations used in Model .................................. 29
Table 17 Summary of Effective Conductivity Equations used in Model ........................ 35
Table 18 Comparison of Velocity and Pressure Maximums for each Case at Ecell=0.7 .. 41
Table 19 Comparison of Maximum Temperatures for each Case at Ecell=0.7 ................ 54
Table 20 Kinetic Models for SOFC MSR and WGS Reactions on Ni Catalysts ............ 74
Table 21 Sensitivity Analysis of Calculated Diffusion Coefficients ............................... 76
Table 22 Polarization Curve Data Table Case 1 to 5 Dampening 0.07% vs Literature
Values .............................................................................................................................. 78
Table 23 Polarization Curve Data Table Case 1 to 5 Dampening 0.05% vs Literature
Values .............................................................................................................................. 78
iv
LIST OF FIGURES
Figure 1 Model Domain..................................................................................................... 7
Figure 2 Relationship between Ideal and True Cell Voltages ........................................ 31
Figure 3 Distribution of Model Mesh Elements for Initial 1/10th of Total Cell Length . 36
Figure 4 Polarization Curves with Model and Experimental data in Literature ............. 38
Figure 5 Comparison Between Case 1 and 4 with Data from Literature......................... 38
Figure 6 Effect of Varying Dampening Factor on Case 1 Polarization Curve ................ 39
Figure 7 Case 1 Inlet Velocity Profile for Ecell=0.7 ........................................................ 40
Figure 8 Case 1 Inlet Velocity Boundary Condition Profiles (m/s) for Ecell=0.7 ............ 40
Figure 9 Case 1 Inlet Pressure Distribution for Ecell=0.7 ................................................ 41
Figure 10 Case 1 Inlet H2 Mole Fractions for Ecell=0.7 ................................................... 43
Figure 11 Case 1 Inlet CO Mole Fractions for Ecell=0.7 .................................................. 43
Figure 12 Case 1 Inlet CO2 Mole Fractions for Ecell=0.7 ................................................ 43
Figure 13 Case 1 Inlet H2O Mole Fractions for Ecell=0.7 ................................................ 44
Figure 14 Case 2 Inlet H2 Mole Fractions for Ecell=0.7 ................................................... 44
Figure 15 Case 2 Inlet CO Mole Fractions for Ecell=0.7 .................................................. 44
Figure 16 Case 2 Inlet CO2 Mole Fractions for Ecell=0.7 ................................................ 44
Figure 17 Case 2 Inlet H2O Mole Fractions for Ecell=0.7 ................................................ 45
Figure 18 Case 3 Inlet H2 Mole Fractions for Ecell=0.7 ................................................... 45
Figure 19 Case 3 Inlet CO Mole Fractions for Ecell=0.7 .................................................. 45
Figure 20 Case 3 Inlet CO2 Mole Fractions for Ecell=0.7 ................................................ 45
Figure 21 Case 3 Inlet H2O Mole Fractions for Ecell=0.7 ................................................ 46
Figure 22 Case 4 Inlet H2 Mole Fractions for Ecell=0.7 ................................................... 46
Figure 23 Case 4 Inlet CO Mole Fractions for Ecell=0.7 .................................................. 46
Figure 24 Case 4 Inlet CO2 Mole Fractions for Ecell=0.7 ................................................ 46
Figure 25 Case 4 Inlet H2O Mole Fractions for Ecell=0.7 ................................................ 47
Figure 26 Case 5 Inlet H2 Mole Fractions for Ecell=0.7 ................................................... 47
Figure 27 Case 5 Inlet CO Mole Fractions for Ecell=0.7 .................................................. 47
Figure 28 Case 5 Inlet CO2 Mole Fractions for Ecell=0.7 ................................................ 47
Figure 29 Case 5 Inlet H2O Mole Fractions for Ecell=0.7 ................................................ 48
v
Figure 30 Case 1 Rate of WGS Reaction (mol/m3-s) for Ecell=0.7 .................................. 48
Figure 31 Case 2 Rate of WGS Reaction (mol/m3-s) for Ecell=0.7 .................................. 49
Figure 32 Case 3 Rate of WGS Reaction (mol/m3-s) for Ecell=0.7 .................................. 49
Figure 33 Case 4 Rate of WGS Reaction (mol/m3-s) for Ecell=0.7 .................................. 49
Figure 34 Case 5 Rate of WGS Reaction (mol/m3-s) for Ecell=0.7 .................................. 49
Figure 35 Case 1 Carbon Formation Activity in the Anode via the Boudouard Reaction
for Ecell=0.7 .................................................................................................................... 50
Figure 36 Case 1 Carbon Formation Activity in the Anode via the Boudouard Reaction
for Ecell=0.95 .................................................................................................................... 51
Figure 37 Case 2 Carbon Formation Activity in the Anode via the Boudouard Reaction
for Ecell=0.95 .................................................................................................................... 51
Figure 38 Case 3 Carbon Formation Activity in the Anode via the Boudouard Reaction
for Ecell=0.95 .................................................................................................................... 51
Figure 39 Case 4 Carbon Formation Activity in the Anode via the Boudouard Reaction
for Ecell=0.95 .................................................................................................................... 51
Figure 40 Case 5 Carbon Formation Activity in the Anode via the Boudouard Reaction
for Ecell=0.95 .................................................................................................................... 52
Figure 41 Case 1 Carbon Formation Activity in the Anode via reaction (72) for
Ecell=0.7 .......................................................................................................................... 52
Figure 42 Case 1 Carbon Formation Activity in the Anode via reaction (72) for
Ecell=0.95 ........................................................................................................................ 52
Figure 43 Case 2 Carbon Formation Activity in the Anode via reaction (72) for
Ecell=0.95 ........................................................................................................................ 53
Figure 44 Case 3 Carbon Formation Activity in the Anode via reaction (72) for
Ecell=0.95 ........................................................................................................................ 53
Figure 45 Case 4 Carbon Formation Activity in the Anode via reaction (72) for
Ecell=0.95 ........................................................................................................................ 53
Figure 46 Case 5 Carbon Formation Activity in the Anode via reaction (72) for
Ecell=0.95 ........................................................................................................................ 53
Figure 47 Case 1 Temperature Distribution for Ecell=0.7 ................................................ 54
Figure 48 Case 1 Temperature Distribution for Ecell=0.6 ................................................ 54
vi
Figure 49 Polarization Curves for Case 1 to Case 5 ........................................................ 55
Figure 50 Case 1 Polarization Curve ............................................................................... 56
Figure 51 Case 2 Polarization Curve ............................................................................... 56
Figure 52 Case 3 Polarization Curve ............................................................................... 57
Figure 53 Case 4 Polarization Curve ............................................................................... 57
Figure 54 Case 5 Polarization Curve ............................................................................... 58
Figure 55 Case 1 Total Local Current density across Anode ERL in y-direction for
Ecell=0.7 ............................................................................................................................ 59
Figure 56 Case 2 Total Local Current density across Anode ERL in y-direction for
Ecell=0.7 ............................................................................................................................ 59
Figure 57 Case 3 Total Local Current density across Anode ERL in y-direction for
Ecell=0.7 ............................................................................................................................ 60
Figure 58 Case 4 Total Local Current density across Anode ERL in y-direction for
Ecell=0.7 ............................................................................................................................ 60
Figure 59 Case 5 Total Local Current density across Anode ERL in y-direction for
Ecell=0.7 ............................................................................................................................ 61
Figure 60 Case 1 Local Current Densities at Anode ERL-Electrolyte Interface for
Ecell=0.7 ............................................................................................................................ 61
Figure 61 Case 2 Local Current Densities at Anode ERL-Electrolyte Interface for
Ecell=0.7 ............................................................................................................................ 62
Figure 62 Case 3 Local Current Densities at Anode ERL-Electrolyte Interface for
Ecell=0.7 ............................................................................................................................ 62
Figure 63 Case 4 Local Current Densities at Anode ERL-Electrolyte Interface for
Ecell=0.7 ............................................................................................................................ 63
Figure 64 Case 5 Local Current Densities at Anode ERL-Electrolyte Interface for
Ecell=0.7 ............................................................................................................................ 63
vii
ABSTRACT
This study considered a planar anode-supported intermediate temperature solid oxide
fuel cell operating on syngas fuel. The effects of varying simulated syngas fuel inlet
compositions on species distribution, temperature distribution, water gas shift reaction
rate, potential for carbon formation and electrochemistry were considered. A 2-D model
was developed containing a composite Ni-YSZ anode, YSZ electrolyte, composite LSMYSZ cathode surrounded by metal interconnects. The domain included separate defined
electrochemical reaction layers on either side of the electrolyte where chemical
reforming and electrochemical reactions simultaneously occurred. Both H2 and CO
electrochemical oxidation was considered along with the internal reforming water gas
shift (WGS) reaction.
The CFD model consists of 5 submodels including the Navier Stokes and continuity
equations for momentum transport, Maxwell-Stefan model considering Knudsen
diffusion for mass transport, energy equation for heat transfer, a chemical reforming
model and an electrochemical model considering distributed charge transfer over the cell
including Butler-Volmer type kinetics.
Resulting polarization curves showed good agreement with experimental data with the
best performance found for higher fuel inlet concentrations of hydrogen and carbon
monoxide. Operating from 0.95 to 0.6V carbon formation in the syngas fueled cell is
unlikely to occur however at higher applied voltages and larger concentrations of
hydrogen and carbon monoxide in the fuel inlet, carbon formation may be possible.
viii
NOMENCLATURE
𝐴𝑣
Electrochemically reactive surface area per unit volume (m2/m3)
𝐶𝑝
Specific Heat Capacity (J/kg-K)
𝑑𝑝𝑜𝑟𝑒
Pore Diameter
𝑀𝑆
𝐷𝑖𝑗
Maxwell-Stefan Diffusivity
DSR
Direct steam reforming chemical reaction
𝐸𝑎
Activation Energy
𝑟𝑒𝑣
𝐸𝑐𝑒𝑙𝑙
Reversible Nernst Open Circuit Cell Potential (V)
𝑟𝑒𝑣
𝐸𝑠,𝑖
Reversible Nernst Half Cell Potential for species i (s = a, c) (V)
𝐸𝑜
Standard Potential (V)
𝐸𝑐𝑒𝑙𝑙
Actual Cell Potential (V)
ERL
Electrochemical Reaction Layer
F
Faraday’s Constant (9.64853 x 104 C/mol)
𝑗
Current Density (A/m2)
𝑗𝑜
Exchange current density
𝑘𝑜
Pre-exponential factor
𝐾𝑝
Equilibrium constant
𝑀i
Molecular weight of species i (g/mol)
MCDR
MSR
Methane carbon dioxide reforming reaction
Methane steam reforming reaction
𝑛𝑒
Number of electrons transferred in rate limiting step
𝑝𝑖
Partial pressure of species i
𝑟̇𝑟𝑥𝑛
Chemical reaction rate (mol/m3-s)
R
Gas Constant (8.3145 J/mol-K)
S/C
Steam to carbon ratio: molar ratio of steam to atomic carbon in fuel
𝜏
Tortuosity
T
Temperature
𝐮
Velocity Vector (m/s)
𝑥𝑖
Mole fraction of species i
ix
𝑦𝑖
WGS
Z
Mass fraction of species i
Water gas shift chemical reaction
Chemical reaction index value [Z=1000/(T(K)-1)]
Greek Letters
α
𝛼𝑐𝑎𝑟𝑏𝑜𝑛
Transfer symmetry coefficient
Carbon activity
𝜀
Porosity
κ
Permeability of porous medium (m2)
𝜆
Thermal Conductivity (W/m-K)
𝜂𝑎𝑐𝑡
Activation Overpotential (V)
𝜎
Conductivity (S/m)
𝜌
Density (kg/m3)
𝜙
Potential (V)
μ
Dynamic viscosity (Pa-s)
𝜐𝑖
Stoichiometric coefficient of species i
Subscripts
𝑎
Anode (subscript)
ba
Anode backing layer (subscript)
bc
Cathode backing layer (subscript)
c
Cathode (subscript)
e
Electrolyte (subscript)
el
Electrical potential (subscript)
io
Ionic potential (subscript)
x
1 Introduction
Fuel cells are promising alternative energy technologies which convert fuel and oxygen
to electricity, water and carbon dioxide. In general, a fuel cell consists of an ion
conducting electrolyte sandwiched in between two porous electrodes. Typically air or
oxygen flows over one of the electrodes (cathode) while hydrogen or a hydrogen
containing fuel flows over the other electrode (anode). In the cathode of the fuel cell, the
oxygen atoms are reduced to oxygen ions which then pass through the electrolyte. Once
they reach the anode, the oxygen ions react with the hydrogen which is oxidized to
produce both water and electrons. The water is carried out of the fuel cell in the anode
flow channel while the electrons are carried through an external circuit back to the
cathode to repeat the process.
There are generally six main types of fuel cells, including polymer electrolyte membrane
fuel cells (PEM), phosphoric acid fuel cells (PAFC), solid oxide fuel cells (SOFC)
alkaline fuel cells (AFC), molten carbonate fuel cells (MCFC), and direct methanol fuel
cells (DMFC). Table 1 below illustrates the general characteristics of each type of fuel
cell.
Table 1 Types of Fuel Cells [1]
Fuel Cell
Temperature (oC)
Applications
Polymer Electrolyte Membrane (PEM)
Phosphoric Acid (PAFC)
60 - 100
175 - 220
Solid Oxide (SOFC)
600 - 1000
Alkaline (AFC)
Molten Carbonate (MCFC)
65 - 220
600 - 650
Direct Methanol (DMFC)
50 - 120
Automotive, Transportation
Distributed generation: Grid support,
cogeneration, stand-alone
Centralized power plant, stand-alone,
cogeneration
Space program, military
Centralized power plant, stand-alone,
cogeneration
Portable small-scale power
In this study, we will focus on solid oxide fuel cells. There are currently several
innovative SOFC products in the market from companies such as Fuel Cell Energy,
Acumentrics, and Bloom Energy. SOFCs are a class of high temperature fuel cells
operating between 600oC to 1000oC which use hydrogen or hydrocarbons as the fuel and
air as the oxidant. This type of fuel cells utilizes porous ceramic electrodes for the anode
1
and cathode, which are separated by a solid ceramic electrolyte. The structure of the
SOFC is commonly referred to as the PEN or positive-electrode/electrolyte/negativeelectrode structure. The two primary configurations of SOFC’s are tubular and planar.
Due to limitations in the performance of tubular SOFC’s, namely that tubular stack
designs have demonstrated low specific power densities, the focus in recent years has
been optimizing the planar design configurations [2].
In the planar type of solid oxide fuel cells evaluated in this study, the general
configuration consists of an interconnect plate, an air/fuel flow channel, the positiveelectrolyte-negative electrode or PEN structure, the alternate air/fuel flow channel and
an alternate interconnect plate. The interconnect plates within a fuel cell stack are
typically fabricated with flow channels on either side, such that only one interconnect
plate is present in the repeating cell units.
The planar type of SOFCs are typically configured in two different ways; either
electrolyte supported or electrode supported. It has been found that under the same
operating conditions, anode supported SOFCs exhibit better performance than
electrolyte supported SOFCs [3]. In the electrode supported fuel cells, the electrode is
the thickest layer in the cell on which all other layers are deposited. Thus for an anode
supported planar SOFC, the anode within the PEN structure provides the structural
support for the unit cell.
Different materials have been studied for use in electrode supported solid oxide fuel
cells. The most common materials utilized today consist of a yttria-stabilized zirconia
(YSZ) electrolyte, and porous ceramic metallic composites (cermets), including
nickel/zirconia (Ni-YSZ) anode and strontium doped LaMnO3 (LSM) mixed with YSZ
composite cathode [4].
SOFC systems can be configured in several different ways according to the approach
taken in the fuel reforming process. In the case where a separate reformer adjacent to the
fuel cell is utilized to extract the hydrogen from the hydrocarbon before feeding it to the
2
fuel cell, this method is called pre-reforming. The other method is feeding the
hydrocarbon directly to the fuel cell where the reformation process takes place on the
catalyst in the anode. This method is called direct internal reforming (DIR). DIR fuel
cells are advantageous over non-DIR fuel cells in that both the fuel reforming and
electrochemical processes occur within the cell, thus a separate reformer is not required
to extract the hydrogen from the hydrocarbon fuel, resulting in less fuel cell powerplant
cost and less overall footprint. They also have increased performance due to the
utilization of the waste heat from the exothermic electrochemical reaction in the
endothermic reforming process. Typical DIR fuel cells can operate at high efficiencies of
50-60%.
Today, the major challenges with DIR SOFCs include material degradation, high cost of
operation, coking, reduced efficiency with higher inlet steam to carbon ratios and sulfur
intolerance.
SOFCs must operate at higher temperatures to both achieve sufficient conversion in the
internal reforming reactions, as well as be able to attain reasonable power densities.
There are several negative aspects to this higher operating temperature, including high
costs of operation, and degradation and cracking in the materials from thermal cycling.
This results in higher maintenance costs as well as a higher cost of material fabrication
due to the need for specialty materials that can survive at the higher temperatures. The
solution in this case would be to operate SOFC’s at lower operating temperatures while
still maintaining high efficiencies. These lower temperature SOFC’s are called
intermediate temperature fuel cells (IT-SOFC). IT-SOFC’s typically operate between
550oC and 800oC (823 K and 1073K).
Another contributor to DIR SOFC material degradation is the non-uniform temperature
distribution across the cell. In a solid oxide fuel cell with direct internal reforming, the
endothermic reforming process generally occurs much faster than the exothermic
electrochemical process. This results in lower temperatures at the anode entrance, large
temperature gradients and thus thermal stress along the cell causing material cracking.
3
Meusinger [5] performed experiments demonstrating that a higher S/C ratio can lower
the temperature gradients in a cell, however using more steam can dramatically decrease
performance. He suggested optimizing the percentage of pre-reforming to lessen the
temperature gradients in the cell. Another approach to lowering the gradients across the
cell is modifying the cell materials by impregnating the anode with copper. This copper
impregnation has been shown to both reduce temperature gradients by lowering the
operating temperature, reducing cost, and reducing carbon formation [6]. A third,
alternative approach to lowering the temperature gradients, which is utilized in this
study, is feeding the fuel cell with a methane-free fuel gas composition, such as syngas,
which eliminates the highly endothermic methane steam reforming reaction.
One of the other challenges with DIR SOFCs is the formation of solid carbon on the
electrode (coking) that blocks or destroys catalyst sites. The general approach to solving
this issue has been to increase the amount of steam in the inlet fuel. However, additional
steam that is added to the inlet fuel reduces the performance by reducing the open circuit
voltage (OCV) in the fuel cell. Fuel stream recycling has been investigated to reduce the
costs of maintaining high steam to carbon ratios, prevent the lowered OCV from the
steam, as well as preventing coking in the anode of the fuel cell [7] [8]. Alternate
catalysts have also been investigated with respect to carbon formation [9]. This study
evaluates the probability of carbon formation on syngas fuel.
To maximize the performance of anode supported DIR SOFCs while minimizing
material degradation and overall cost, the system performance including the inlet species
concentrations, inlet conditions, cell flow configurations and the thermal management
within the fuel cell along with the material optimizations mentioned previously must
also be optimized. Among the types of modeling utilized to simulate the SOFC single
cell level conditions for optimization, generally the more predominant models are of the
type where the electrochemical reactions are defined to occur at the electrode-electrolyte
interfaces which Hussain et al. [10] refers to as macro-level models. The other primary
approach, micro-level type models assume the electrochemical reactions occur
throughout the electrode and typically focus on only one electrode. However, as Hussain
4
et al. notes, incorporating the two types of models together enhances the predictive
capability of a cell level study. To incorporate the two approaches, distinctive
electrochemically reactive layers are introduced into the model between the bulk
electrode and the electrolyte in this study.
Of the researchers that have considered this novel approach using distinct
electrochemical reactive layers, Hussain et. al. [10] developed a distributed charge
transfer numerical model to predict the electrochemical performance characteristics of a
DIR SOFC utilizing a distributed charge transfer model not considering CO oxidation.
Ho et al. [11] developed a numerical model to determine the electrochemical
performance and temperature distribution of an anode supported IT-DIR SOFC using a
modified Nernst-Planck equation including internal reforming and CO oxidation.
Anderssen [12] developed a 2-D COMSOL CFD model of a IT DIR SOFC including
internal reforming and CO oxidation comparing the effect of varying inlet fuel and air
compositions, utilizations and inlet velocities on co-flow cell performance. Jeon [13]
developed a 2-D IT SOFC CFD model examining the effect of parameters on
performance and temperature distribution including distinct electrochemical reaction
layers using only H2 fuel.
It has been found that the overall electrochemical reaction rate can vary up to 50% with
the oxidation of CO when compared with only the oxidation of hydrogen [14] therefore
for model accuracy both hydrogen oxidation and CO oxidation should be included in a
cell level study where anode species composition could contain notable amounts of CO,
such as in pre-reformed hydrocarbon feeds, syngas feeds, or fuel cells with internal
reforming. Increasingly, CO oxidation is being included in SOFC modeling. Of the
recent models that consider CO oxidation not already mentioned; Suwanwarangkul [15]
developed a multi-physics COMSOL model based on experimental button cell data
analyzing performance on syngas fuel and thermodynamic potential for carbon
formation with unique exchange current density kinetics. Andreassi [16] used COMSOL
and developed an experimentally validated 3-D model for a CO/H2/CO2/H2O fed planar
SOFC including WGS kinetics. Using electrochemical kinetics similar to that of
5
Andreassi, Razbani [17] utilized COMSOL to develop a cross-flow model validated
against experimental data on a electrolyte supported planar SOFC stack fed with
H2/CO2. Iwai et al developed a numerical model of a DIR SOFC using an equivalent
circuit approach involving a volume averaging method to examine the electrochemical
performance and thermal distribution considering the MSR, WGS and MCDR reaction
with high methane and CO2 content fuel [18]. Park et al developed a 3D numerical
model of a DIR SOFC examining the effect of inlet species concentrations and S/C ratio
on chemical and electrochemical reactions and cell performance [14]. Ni et al developed
a 2D model of a DIR SOFC examining chemical kinetics approaches and operating
parameters on performance [19]. Nikooyeh developed a 3-D CFD model of a DIR
SOFC examining carbon formation and recycling of gas exhaust [8].
2 Methodology
2.1 Domain and Physical Parameters
This study considered a 2D model of a planar anode-supported IT-DIR SOFC with
composite Ni-YSZ anode and composite LSM-YSZ cathode. In the 2-D single cell
model shown in Figure 1, there are a total of nine distinct layers, each having a total
length of 100 mm and heights as defined in Table 2.
In this particular model, the active catalyst electrochemical reaction layers (ERL) were
treated as a separate layer from the electrode to replicate the location of the
electrochemically active zone at the boundaries of the electrode and electrolyte layers.
This defined separate layer approach is a more accurate simulation than the assumption
of surface electrochemical reactions only, as it has been shown that the electrochemical
reactions occur within the electrode at a distance of 10 to 50µm away from the
electrode-electrolyte interface [11] [10]. The materials and properties assumed in the
model are listed in the following tables.
6
Cell length
Cell height
Interconnect Height
Fuel channel height
Anode Backing Layer Height
Anode ERL Layer Height
Table 2 Cell Dimensions (mm)
100
Air channel height
3.31
Cathode Backing Layer Height
0.5
Cathode ERL Layer Height
0.6
Electrolyte Height
0.6
0.03
1.0
0.05
0.01
0.02
Table 3 Cell Materials
Anode and Cathode Interconnect
Stainless Steel
Anode Electrode and Anode ERL Layer
Ni-YSZ (Nickel - Yttria Stabilized Zirconia)
Electrolyte
YSZ (Yttria Stabilized Zirconia)
LSM-YSZ (Strontium doped Lanthanum
Cathode Electrode and Cathode ERL Layer
Manganite – Yttria Stabilized Zirconia)
Figure 1 Model Domain
7
The following table details the physical properties, and electrochemical/thermal
parameters assumed for the cell. More research has been performed on Ni-YSZ anode
characteristics than LSM-YSZ cathode characteristics, therefore in the cases where
cathode properties or parameters were unavailable, the corresponding anode values were
utilized.
Table 4 Base Case Physical Properties and Parameters
Permeability (m )
Porosity
Pore Diameter (µm)
Anode
2.42 x 10 -14
0.489
0.971
Cathode
2.54 x 10 -14
0.515
1
Electronic/Ionic/Pore Tortuosity
7.53, 8.48, 1.80
7.53, 3.4, 1.80
Electronic/Ionic Volume Fraction
0.257, 0.254
0.232, 0.253
Electronic/Ionic Reactive Surface Area
per Unit Volume (m2/m3)
Solid Thermal Conductivity (W/m-K)
Solid Specific Heat Capacity (J/kg-K)
Solid Density (kg/m3)
3.97x10 6 , 7.93x10 6
3.97x10 6 , 7.93x10 6
[23]
11
450
3310
Electrolyte
2.7
470
5160
6
430
3030
Interconnect
20
550
3030
[12]
[12]
[12]
2
Thermal Conductivity (W/m-K)
Specific Heat Capacity (J/kg-K)
Solid Density (kg/m3)
[20] [21]
[20]
[20] [22]
[21]
[20] [22]
[21]
[12]
[12]
[12]
2.2 Operating Conditions
The model operating conditions are shown in Table 5 with the varied fuel inlet
compositions shown in Table 6. The inlet fuel includes low amounts of CH4 with Case 1
through 3 representing typical ranges for syngas fuels.
Table 5 Model Operating Conditions
Inlet Temperature (K)
Cathode Inlet Velocity (m/s)
Anode Inlet Velocity (m/s)
Outlet Pressure (atm)
1023
6.5
0.5
1.0
Anode Fuel Feed xi
Cathode Air Feed xi
Operating Voltage (V)
8
Varies
.21 O2 .79 N2
0.6 to 1.0
Table 6 Simulated Fuel Feed Mole Fractions
Case
1
2
3
4
5
H2
0.30
0.30
0.20
0.30
0.30
H2O
0.07
0.17
0.27
0.07
0.07
CO
0.50
0.40
0.40
0.40
0.40
CO2
0.10
0.10
0.10
0.10
0.20
CH4
0.01
0.01
0.01
0.01
0.01
N2
0.02
0.02
0.02
0.12
0.02
2.3 CFD Model Overview
The commercially available software COMSOL was used to model the domain. The
domain parameters are shown in Table 5. The 2-D computational fluid dynamics (CFD)
model consists of conservation equations for mass, momentum, species, charge and
energy. Using the Free and Porous Media Flow Module, Navier-Stokes equations are
utilized to model the flow in the anode and cathode flow channels, and the Brinkman
equations are utilized to model the flow in the porous electrodes. For the mass balances,
the Transport of Concentrated Species Module was used. It includes Maxwell-Stefan
diffusion, where species transfer and kinetic rate equations were considered for both the
chemical and electrochemical reactions. The energy balance was incorporated through
the use of the Heat Transfer in Fluids Module. For the electronic and ionic charge
balance, the appropriate distributed charge transfer equations were entered manually into
the mathematics module using Poisson’s equations. All other definitions and equations
utilized are manually entered as defined variables applied into the model. More details
on all the equations utilized can be found in the modeling sections of this paper.
3 Momentum Model
The Free and Porous Media Flow module was selected to model the momentum balance
and calculate the velocity fields and pressure gradients across the cell in the fuel cell
flow fields and electrodes at steady state. This program module includes the
functionality to model systems with both free and porous media flow.
9
3.1 General Equations
For the open flow channels in the cell, including the fuel and air flow channels, the
following Continuity and Navier Stokes Equations were utilized considering
compressible flow and steady state conditions.
∇ ∙ (ρ𝐮) = 0
(1)
2
ρ𝐮 ∙ ∇𝐮 = ∇ ∙ [−p𝐈 + μ ((∇𝐮 + (∇𝐮)T ) − μ(∇ ∙ 𝐮)𝐈)] + 𝐅
3
(2)
For the flow in the porous electrodes (and corresponding electrode reaction layers) the
Stokes-Brinkman equations were utilized, which neglect the initial term in the Brinkman
equations due to very low Reynolds number.
∇ ∙ (ρ𝐮) = S
(3)
μ
μ
2μ
𝐮 ( + S) = ∇ ∙ [−p𝐈 + ((∇𝐮 + (∇𝐮)T ) − (∇ ∙ 𝐮)𝐈)] + 𝐅
κ
𝜀
3𝜀
(4)
In these equations S is the mass source term (kg/m3-s), F is the volume force vector, μ is
viscosity, κ is permeability, u is the velocity vector, ρ is density, I is the unit matrix.
The boundary conditions utilized in the model included: no slip conditions at the walls,
specified inlet velocities at the anode and cathode and outlet pressure of 101325 Pa.
3.2 Density and Viscosity
The density of the gases is dependent on temperature as defined in the Transport of
Concentrated Species module, and is determined from the ideal gas model. The dynamic
viscosity μ of a mixture is dependent on both the temperature and mixture composition.
To calculate the dynamic viscosity of these low pressure mixtures, there are several
methods available varying in complexity [24]. For this study, thermodynamic data in
10
Table 7 was utilized in the following equations as a combination of the Wilke and
Herning & Zipperer methods.
𝜇𝑖 = 1𝑥10−7 [𝑎0 + 𝑎1 (𝑇⁄1000) + 𝑎2 (𝑇⁄1000)2 + 𝑎3 (𝑇⁄1000)3 + 𝑎4 (𝑇⁄1000)4
(5)
+ 𝑎5 (𝑇⁄1000)5 + 𝑎6 (𝑇⁄1000)6 ]
𝑁
𝑥𝑖 𝜇𝑖
𝑛
∑𝑗=1 𝑥𝑗 𝜃𝑖𝑗
(6)
𝜃𝑖𝑗 = (𝑀𝑗 /𝑀𝑖 )1/2 = 𝜃𝑗𝑖−1
(7)
𝜇𝑚𝑖𝑥𝑡𝑢𝑟𝑒 = ∑
𝑖=1
In these equations, T is in Kelvin, 𝜇𝑖 is the species dynamic viscosity in Pa-s (conversion
made by multiplying 1𝑥10−7), 𝑥𝑖 is the mole fraction of species i, and 𝑀𝑖 is the
molecular weight of species i. For the binary mixture in the cathode (O2 and N2) the
following equations result:
𝜇𝑐𝑎𝑡ℎ𝑜𝑑𝑒 =
𝑥𝑂2 μ𝑂2
𝑥𝑁2 μ𝑁2
+
𝑥𝑂2 + 𝑥𝑁2 𝜃𝑂2,𝑁2 𝑥𝑁2 + 𝑥𝑂2 𝜃𝑁2,𝑂2
𝜃𝑂2,𝑁2 = (𝑀𝑁2 /𝑀𝑂2 )1/2
𝜃𝑁2,𝑂2 =
1
(8)
(9)
(10)
𝜃𝑂2,𝑁2
In the anode, the equation set for dynamic viscosity becomes significantly more
complicated due to the greater number of species. In total there are 6 species including:
CH4, H2, H2O, CO, CO2 and N2. For a 6 species mixture equation (7) is combined with
the following definitions.
𝜇𝑎𝑛𝑜𝑑𝑒 =
𝑥1 𝜇1 𝑥2 𝜇2 𝑥3 𝜇3 𝑥4 𝜇4 𝑥5 𝜇5 𝑥6 𝜇6
+
+
+
+
+
𝛽1
𝛽2
𝛽3
𝛽4
𝛽5
𝛽6
𝛽1 = 𝑥1 + 𝑥2 𝜃12 + 𝑥3 𝜃13 + 𝑥4 𝜃14 + 𝑥5 𝜃15 + 𝑥6 𝜃16
𝛽2 =
𝑥1
+ 𝑥2 + 𝑥3 𝜃23 + 𝑥4 𝜃24 + 𝑥5 𝜃25 + 𝑥6 𝜃26
𝜃12
11
(11)
(12)
(13)
𝑥1
𝑥2
+
+ 𝑥3 + 𝑥4 𝜃34 + 𝑥5 𝜃35 + 𝑥6 𝜃36
𝜃13 𝜃23
x1
x2
x3
β4 =
+
+
+ x4 + x5 𝜃45 + 𝑥6 𝜃46
𝜃14 𝜃24 𝜃34
x1
x2
x3
x4
β5 =
+
+
+
+ x5 + 𝑥6 𝜃56
𝜃15 𝜃25 𝜃35 𝜃45
x1
x2
x3
x4
x5
β6 =
+
+
+
+
+ 𝑥6
𝜃16 𝜃26 𝜃36 𝜃46 𝜃56
(14)
𝛽3 =
CH4
H2O
CO2
CO
H2
N2
O2
a0
-9.9989
-6.7541
-20.434
-4.9137
15.553
1.2719
-1.6918
(15)
(16)
(17)
Table 7 Species Dynamic Viscosity Coefficients [25]
a1
a2
a3
a4
a5
529.37
-543.82
548.11
-367.06
140.48
244.93
419.50
-522.38
348.12
-126.96
680.07
-432.49
244.22
-85.929
14.450
793.65
875.90
883.75
-572.14
208.42
299.78
-244.34
249.41
-167.51
62.966
771.45
-809.20
832.47
-553.93
206.15
889.75
-892.79
905.98
-598.36
221.64
a6
-22.920
19.591
-0.4564
-32.298
-9.9892
-32.430
-34.754
3.3 Microstructural Properties
In determining the momentum balance for the porous electrodes, the microstructure
material properties need to be defined. The porosity and the permeability are important
values
typically determined experimentally along with
other
microstructural
characteristics for the particular material. Cell performance, namely electrical
conductivity and the effective gas diffusion within an electrode, depends on the pore
structure within the material. To determine the permeability, the Carman-Kozeny
correlation, which is derived from Darcy’s Law is used (assuming laminar flow with Re
< 2300). The general form for the Carman-Kozeny equation is:
𝜅=
𝜀 3 𝑑𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 2
c(1 − 𝜀 )2
(18)
The Carman-Kozeny equation can be used with the constant 𝑐 set equal to 180 or 150
which are both empirical values commonly used for particles assumed to be spherical in
shape. Zhu et al [26] proposed the constant 𝑐 may also be further defined in terms of
tortuosity where 𝑐 = 𝜏𝑐𝑜 and 𝑐𝑜 is a shape factor 𝑐𝑜 = 72. However the resulting
12
calculated permeability was one to three orders of magnitude smaller than the reported
values by Kishimoto [20], who calculated the permeability of Ni-YSZ cermet anodes
based on the experimentally determined microstructural characteristics of pore volume
(porosity), pore surface to volume ratio (S/V)pore and pore tortuosity, as shown in the
equation below. The results of Kishimoto closely matched the microstructural
characteristics determined via the dual beam FIB-SEM experiment by Iwai et al [22]. A
summary of microstructural parameters in the literature, averaged where appropriate, are
presented in Table 8. The calculated permeability by Kishimoto using (S/V)pore =
4.33 x 106 and the following equation is assessed in this study.
𝜅=
𝑉𝑝𝑜𝑟𝑒
(19)
6𝜏𝑝𝑜𝑟𝑒 (S/V)2pore
Table 8 Ni-YSZ Anode Microstructural Characteristics in Literature
Source
𝜅 (𝑚2 )
*Kishimoto
[20]
2.415 x 10 -15**
𝑉𝑝𝑜𝑟𝑒
𝑑𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑑𝑝𝑜𝑟𝑒
𝑉𝑒𝑙
𝑉𝑖𝑜
𝜏𝑝𝑜𝑟𝑒
𝜏𝑒𝑙
𝜏𝑖𝑜
0.489
0.257
0.254
1.80
8.13
7.11
0.489
0.257
0.254
1.96
6.93
9.85
0.35
0.23
0.42
4.5
-
-
0.30
0.28
0.42
-
10
10
(μ𝑚)
(μ𝑚)
or 𝜀
-
0.971
*Iwai
[22]
Zhu
4.510 x 10-16 to
0.1 –
[26]
3.132 x 10-18
1.2
Anderssen
1.76 x 10 -11
0.34
[12]
*Indicates experimentally determined values
**Calculated parameter
Two approaches are available for calculating the pore diameter if not available from
experiment. The first assumes that the mean pore diameter is equivalent to the hydraulic
diameter [20]. The second calculates the pore diameter as a function of the electrode
porosity and particle diameter 𝑑𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 [13].
4
(𝑆/𝑉)𝑝𝑜𝑟𝑒
(20)
2 𝜀
𝑑
3 1 − 𝜀 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒
(21)
𝑑𝑝𝑜𝑟𝑒 ≈ 𝑑ℎ =
𝑑𝑝𝑜𝑟𝑒 =
13
4 Mass Transfer Model
The mass transport in the SOFC is due to Maxwell-Stefan and Knudsen diffusion along
with convection. The consumption and generation of species in the cell is implemented
using chemical and electrochemical kinetic rate expressions as source terms in the mass
transport equations. The Transport of Concentrated Species module was used in
modeling the mass transport through the cell.
4.1 General Equations
The following mass transport equations are utilized for a steady state system to model
species transport in the fuel cell flow fields and electrodes.
𝑁
̃𝑖𝑗 𝒅𝑗 + 𝐷𝑖𝑇
𝜌𝑦𝑖 (∇ ∙ 𝒖) − ∇ ∙ (𝜌𝑦𝑖 ∑ 𝐷
𝑗
𝒅𝑗= ∇𝑥𝑗 +
1
𝑝𝑎𝑏𝑠
𝛻𝑇
) = 𝑅𝑖
𝑇
⌈(𝑥𝑗 − 𝑦𝑗 )∇𝑝𝑎𝑏𝑠 ⌉
(22)
(23)
̃ ij are the Fick diffusivity values (m2/s) calculated from the
In the equations above, D
T
2
Maxwell-Stefan diffusivity matrix values DMS
ij in (m /s) as shown in reference [27], Di
are thermal diffusion coefficients (kg/m-s), and R i represents the species source term
defined by the kinetic rate expressions for species generation or consumption in the
chemical/electrochemical reactions (kg/(m3-s). The subscript i indicates each unique
species in consideration. For the boundary conditions in the model, no mass flux was
assumed where there was no flow of species (i.e. outside the cell other than in the flow
channels and electrodes), and the inflow species concentrations were predefined mass
fractions. The velocity and pressure values were derived from the momentum balance,
and to determine the mixture density the gases were assumed to be ideal.
14
4.2 Maxwell-Stefan Diffusivity
2
To calculate the Maxwell-Stefan diffusivity values DMS
ij (m /s) for the species present in
the non-porous flow fields, there are two types of equations generally used. The first
type utilizes the Chapman-Enskog theory combined with Lennard-Jones parameters [11]
[14], while the second type utilizes the Fuller expression shown below [25]. The values
Vi and Vj are specific diffusion volumes calculated in the Fuller method, 𝑇 is in K and p
is in Pa.
𝑀𝑆
𝐷𝑖𝑗
=
1.43𝑥10−2 𝑇 1.75
1/2
𝑝𝑀𝑖𝑗 [𝑉𝑖1
𝑀𝑖𝑗 =
⁄3
⁄
(24)
2
+ 𝑉𝑗 1 3 ]
2
1
1
+
𝑀𝑖 𝑀𝑗
(25)
Table 9 Fuller Diffusion Volume [25]
CH4
H2O
CO2
CO
H2
Fuller Diffusion
Volume
25.14
13.1
26.7
18.0
6.12
N2
O2
18.5
16.3
The above equation is practical for use in the flow fields, however another approach
utilizing the Dusty Gas Model must be taken for diffusion in porous media. A new
diffusivity value is introduced called the effective diffusivity Deff
ij , which combines the
standard binary diffusivity with the Knudsen diffusivity DKn
ij and introduces the effects
of pore characteristics including tortuosity and porosity [12] [10] [23].
𝑒𝑓𝑓
𝐷𝑖𝑗
=
𝜀
1
1
( 𝑀𝑆 + 𝐾𝑛 )
𝜏𝑝𝑜𝑟𝑒 𝐷𝑖𝑗
𝐷𝑖
−1
=
𝜀
𝑀𝑆 𝐾𝑛
𝐷𝑖𝑗
𝐷𝑖𝑗
𝐾𝑛
𝜏𝑝𝑜𝑟𝑒 𝐷𝑖𝑗
+
(26)
𝑀𝑆
𝐷𝑖𝑗
2
8𝑅𝑇
𝑇
𝐾𝑛
𝐷𝑖𝑗
= 𝑟𝑝𝑜𝑟𝑒 𝑥10−4 √
= 48.5𝑥10−4 𝑑𝑝𝑜𝑟𝑒 √
̅
̅
3
𝜋𝑀𝑖𝑗
𝑀𝑖𝑗
(27)
̅𝑖𝑗 = (𝑀𝑖 + 𝑀𝑗 )/2
𝑀
(28)
15
In the previous equations, the pore diameter 𝑑𝑝𝑜𝑟𝑒 is in meters, the temperature of the
2
diffusing medium T is in K and DKn
ij is m /s.
The thermal diffusion coefficient DTi is not typically included in cell sized modeling.
There has been much work in the past to determine the thermal diffusion coefficient and
predictive equations for binary mixtures and some work on ternary mixtures. However
data for multicomponent mixtures, specifically for the gaseous mixture in the anode, is
not currently available. Based on this, the thermal diffusion coefficient will not be
considered in this study. Thus, the mass transfer equation reduces to the following.
𝑁
(29)
̃𝑖𝑗 𝒅𝑗 ) = 𝑅𝑖
𝜌𝑦𝑖 (∇ ∙ 𝒖) − ∇ ∙ (𝜌𝑦𝑖 ∑ 𝐷
𝑗
5 Heat Transfer Model
To model the heat transfer within the cell, The Heat Transfer Modules in COMSOL
were utilized. Depending on the layer considered, different forms of the energy equation
for each type of domain were considered.
5.1 General Equations
5.1.1. Flow Fields
For the heat transfer in the gas flow fields the following form of the energy equation was
used:
𝜌𝐶𝑝 𝒖∇𝑇 − ∇(𝜆∇𝑇) = 𝑄
(30)
where 𝐶𝑝 is the specific heat capacity (J/kg-K) at constant pressure, 𝜆 is the thermal
conductivity (W/m-K), and 𝑄 is the heat generation or source term discussed in Section
5.2. To determine the specific heat capacity of the fluid mixtures, ideal gases were
16
assumed such that the heat capacities were functions of temperature only per the
following equations [25]
𝐶𝑝,𝑖 =
1000
[𝑏0 + 𝑏1 (𝑇⁄1000) + 𝑏2 (𝑇⁄1000)2 + 𝑏3 (𝑇⁄1000)3 + 𝑏4 (𝑇⁄1000)4
𝑀𝑖
(31)
+ 𝑏5 (𝑇⁄1000)5 + 𝑏6 (𝑇⁄1000)6 ]
𝑁
(32)
𝐶𝑝,𝑚𝑖𝑥𝑡𝑢𝑟𝑒 = ∑ 𝑥𝑖 𝐶𝑝,𝑖
𝑖=1
In these equations 𝐶𝑝 is the specific heat (J/kg-K), T is temperature in K, and 𝑥𝑖 is the
species mole fraction. Applying this equation to the cathode of the cell yields the
following equation
𝐶𝑝,𝑐𝑎𝑡ℎ𝑜𝑑𝑒 = 𝑥𝑂2 𝐶𝑝,𝑂2 + 𝑥𝑁2 𝐶𝑝,𝑁2
(33)
In the anode we must account for the additional species present, as shown in the
following equation
𝐶𝑝,𝑎𝑛𝑜𝑑𝑒 = 𝑥𝐶𝐻4 𝐶𝑝,𝐶𝐻4 + 𝑥𝐻2𝑂 𝐶𝑝,𝐻2𝑂 + 𝑥𝐶𝑂2 𝐶𝑝,𝐶𝑂2 + 𝑥𝐶𝑂 𝐶𝑝,𝐶𝑂 + 𝑥𝐻2 𝐶𝑝,𝐻2 + 𝑥𝑁2 𝐶𝑝,𝑁2
CH4
H2O
CO2
CO
H2
N2
O2
b0
47.964
37.373
4.3669
30.429
21.157
29.027
34.850
Table 10 Species Heat Capacity Coefficients [25]
b1
b2
b3
b4
-178.59
712.55
-1068.7
856.93
-41.205
146.01
-217.08
181.54
204.60
-471.33
657.88
-519.9
-8.1781
5.2062
41.974
-66.346
56.036
-150.55
199.29
-136.15
4.8987
-38.040
105.17
-113.56
-57.975
203.68
-300.37
231.72
b5
-358.75
-79.409
214.58
37.756
46.903
55.554
-91.821
(34)
b6
61.321
14.015
-35.992
-7.6538
-6.4725
-10.350
14.776
To determine the thermal conductivity of the fluid mixtures, the method of Wassiljewa
was utilized [24] with the Mason and Saxena modification as suggested by Todd and
Young [25]. It is interesting to note that equations similar to those used to calculate
thermal conductivity in this study are used by Wilke in an alternate calculation for
17
mixture viscosity not used in this model [24]. Using the equations of Mason and Saxena,
the following applies for the thermal conductivity of the fluids
𝜆𝑖 = 𝑐0 + 𝑐1 (𝑇⁄1000) + 𝑐2 (𝑇⁄1000)2 + 𝑐3 (𝑇⁄1000)3 + 𝑐4 (𝑇⁄1000)4
(35)
+ 𝑐5 (𝑇⁄1000)5 + 𝑐6 (𝑇⁄1000)6
𝑁
𝜆𝑚𝑖𝑥𝑡𝑢𝑟𝑒
(36)
𝑥𝑖 𝜆𝑖
=∑ 𝑛
∑𝑗=1 𝑥𝑗 ∅𝑖𝑗
𝑖=1
1/2
∅𝑖𝑗 =
[1 + (𝜇𝑖 / 𝜇𝑗 )
1/4 2
(𝑀𝑖 /𝑀𝑗 )
(37)
]
1/2
[8(1 + 𝑀𝑖 /𝑀𝑗 )]
∅𝑗𝑖 = ∅𝑖𝑗 (𝜇𝑗 𝑀𝑖 /𝜇𝑖 𝑀𝑗 )
(38)
where 𝜆 is in W/m-K, and 𝜇𝑖 is the species dynamic viscosity in µPoise. Applying these
equations to the cathode fluid mixture yields the following equation set for the mixture
thermal conductivity,
𝜆𝑐𝑎𝑡ℎ𝑜𝑑𝑒 =
∅𝑂2,𝑁2 =
𝑥𝑂2 𝜆𝑂2
𝑥𝑁2 𝜆𝑁2
+
𝑥𝑂2 +𝑥𝑁2 ∅𝑂2,𝑁2 𝑥𝑂2 ∅𝑁2,𝑂2 + 𝑥𝑁2
[1 + (𝜇𝑂2 / 𝜇𝑁2 )1/2 (𝑀𝑂2 /𝑀𝑁2 )1/4 ]
[8(1 + 𝑀𝑂2 /𝑀𝑁2 )]1/2
2
∅𝑁2,𝑂2 = ∅𝑂2,𝑁2 (𝜇𝑁2 𝑀𝑂2 /𝜇𝑂2 𝑀𝑁2 )
(39)
(40)
(41)
In the anode the additional species in the gas mixture must be accounted for. In total
there are 6 possible species including: CH4, H2, H2O, CO, CO2, and N2. For a 6 species
mixture, equation (37) is combined with the following definitions to determine mixture
thermal conductivity,
𝜆𝑎𝑛𝑜𝑑𝑒 =
𝑥1 𝜆1 𝑥2 𝜆2 𝑥3 𝜆3 𝑥4 𝜆4 𝑥5 𝜆5 𝑥6 𝜆6
+
+
+
+
+
𝛿1
𝛿2
𝛿3
𝛿4
𝛿5
𝛿6
𝛿1 = 𝑥1 + 𝑥2 ∅12 + 𝑥3 ∅13 + 𝑥4 ∅14 + 𝑥5 ∅15
𝛿2 = 𝑥1 ∅12
𝜇2 𝑀1
+ 𝑥2 + 𝑥3 ∅23 + 𝑥4 ∅24 + 𝑥5 ∅25
𝜇1 𝑀2
18
(42)
(43)
(44)
𝜇3 𝑀1
𝜇3 𝑀2
+ 𝑥2 ∅23
+ 𝑥3 + 𝑥4 ∅34 + 𝑥5 ∅35
𝜇1 𝑀3
𝜇2 𝑀3
(45)
𝜇4 𝑀1
𝜇4 𝑀2
𝜇4 𝑀3
+ 𝑥2 ∅24
+ 𝑥3 ∅34
+ x4 + x5 ∅45
𝜇1 𝑀4
𝜇2 𝑀4
𝜇3 𝑀4
(46)
𝜇5 𝑀1
𝜇5 𝑀2
𝜇5 𝑀3
𝜇5 𝑀4
+ 𝑥2 ∅25
+ 𝑥3 ∅35
+ 𝑥4 ∅45
+ x5
𝜇1 𝑀5
𝜇2 𝑀5
𝜇3 𝑀5
𝜇4 𝑀5
(47)
𝜇6 𝑀1
𝜇6 𝑀2
𝜇6 𝑀3
𝜇6 𝑀4
𝜇6 𝑀5
+ 𝑥2 ∅26
+ 𝑥3 ∅36
+ 𝑥4 ∅46
+ x5
+ x6
𝜇1 𝑀6
𝜇2 𝑀6
𝜇3 𝑀6
𝜇4 𝑀6
𝜇4 𝑀5
(48)
𝛿3 = 𝑥1 ∅13
δ4 = 𝑥1 ∅14
δ5 = 𝑥1 ∅15
δ6 = 𝑥1 ∅16
CH4
H2O
CO2
CO
H2
N2
O2
c0
0.4796
2.0103
2.8888
-0.2815
1.5040
0.3216
-0.1857
Table 11 Species Thermal Conductivity Coefficients [25]
c1
c2
c3
c4
c5
1.8732
37.413
-47.440
38.251
-17.283
-7.9139
35.922
-41.390
35.993
-18.974
-27.018
129.65
-233.29
216.83
-101.12
13.999
-23.186
36.018
-30.818
13.379
62.892
-47.190
47.763
-31.939
11.972
14.810
25.473
38.837
32.133
13.493
11.118
-7.3734
6.7130
-4.1797
1.4910
c6
3.2774
4.1531
18.698
-2.3224
-1.8954
2.2741
-0.2278
5.1.2. Electrodes, Electrolyte and Interconnects
In the electrodes (both the backing layers and ERLs) a modified version of the heat
equation is utilized, which introduces the values of effective thermal conductivity and
effective specific heat capacity. These values are introduced to account for the electrode
porosity [10] [19] [12]. The heat generation source term 𝑄 is discussed in Section 5.2
𝑒𝑓𝑓
𝜌𝐶𝑝 𝒖∇𝑇 − ∇(𝜆𝑒𝑓𝑓 ∇𝑇) = 𝑄
(49)
𝜆𝑒𝑓𝑓 = 𝜀𝜆𝑓𝑙𝑢𝑖𝑑 + (1 − 𝜀)𝜆𝑠𝑜𝑙𝑖𝑑
(50)
𝑒𝑓𝑓
(51)
𝐶𝑝
= 𝜀𝐶𝑝,𝑓𝑙𝑢𝑖𝑑 + (1 − 𝜀)𝐶𝑝,𝑠𝑜𝑙𝑖𝑑
The subscript “fluid” is the calculated thermal conductivity and specific heat capacity of
the fluid mixture in the anode or cathode electrode using the methods described in the
previous section. The subscript “solid” indicates the thermal conductivity or specific
heat capacity of the solid phase of the anode or cathode provided in specified model
parameters.
19
For the electrolyte and interconnects, the following form of the heat equation
considering conduction heat transfer is used where the thermal conductivity is a predefined value taken from the literature.
−∇(𝜆∇𝑇) = 𝑄
(52)
5.2 Heat Generation Source Terms
There are various sources of heat generation and consumption in the solid oxide fuel
cell. For a methane fed SOFC the primary influences on the cell temperature gradient
occur due to the methane steam reaction and the electrochemical reactions. A summary
of the relative contributions of each type of heat source/sink found in the fuel cell was
presented in the ASME report by M. Navasa [28] and is reproduced here for reference.
Table 12 Types of SOFC Heat Sources [28]
Fuel Cell
Type
Relative % Contribution
MSR Reaction
Consumption
27
WGS Reaction
Electrochemical Reactions
Concentration Polarization
Generation
Generation
Generation
6
47
<1
Activation Polarization
Ohmic Polarization
Generation
Generation
16
3
5.2.1. Heat Generated by Reactions
The general equation form for the heat generated by the chemical reactions modeled in
this study shown below. In this equation ∆𝐻298 is in J/mol.
𝑟𝑥𝑛
𝑟𝑥𝑛
𝑄 = ∑ − (∆𝐻298
∗ 𝑟̇ 𝑟𝑥𝑛 )
(53)
𝑗
The heat generated from the electrochemical reactions applies only in the
electrochemically reactive layers (ERLs) of the cell. The following is the general form
20
for these equations applied in the anode ERL of the cell for the H2 and CO oxidation
reactions [8]. As the current density is calculated based on unit area it is multiplied by
the reactive surface area 𝐴𝑣 (m2/m3) to obtain the volumetric current density.
𝑄 = ∑ 𝑗𝑠,𝑖 𝐴𝑣 (
𝑖
−∆𝐻𝑖
− 𝐸𝑐𝑒𝑙𝑙 )
𝑛𝑒,𝑖 𝐹
∆𝐻𝐻2𝑜𝑥 = −248.42 𝑘𝐽/𝑚𝑜𝑙 ∆𝐻𝐶𝑂𝑜𝑥 = −282.47 𝑘𝐽/𝑚𝑜𝑙
(54)
(55)
5.2.2. Heat Generation from Polarizations
The heat generation due to activation polarizations in the cell can be calculated using the
common equation shown below and is applied to the appropriate layers where s = a or c
(anode or cathode).
𝑄 = 𝜂𝑎𝑐𝑡,𝑠 𝑗𝑠 𝐴𝑣
(56)
Joule heating or heat generation due to ohmic polarizations in SOFC modeling is the
heat generated from the resistance to ion or electron flow in the cell. As shown in Table
12, the contribution to the overall heat generation in the cell from joule heating is quite
low on the order of 3% and was thus neglected in this study.
Additionally, due to the low relative contribution of heat generation due to concentration
polarizations, < 1%, this source term was neglected as well. The summary of heat source
terms applied to their respective domains is shown in the table below.
Table 13 Summary of Heat Source Equations used in Model
21
Anode Flow Field
𝑊𝐺𝑆
𝑄 = (−∆𝐻298
∗ 𝑟̇ 𝑊𝐺𝑆 )
(57)
Anode Backing Layer
𝑊𝐺𝑆
𝑄 = (−∆𝐻298
∗ 𝑟̇ 𝑊𝐺𝑆 )
(58)
Anode ERL
−∆𝐻𝐻2
𝑊𝐺𝑆
𝑄 = (−∆𝐻298
∗ 𝑟̇ 𝑊𝐺𝑆 ) + 𝑗𝐻2 𝐴𝑣 (
− 𝐸𝑐𝑒𝑙𝑙 )
2𝐹
−∆𝐻𝐶𝑂
+ 𝑗𝐶𝑂 𝐴𝑣 (
− 𝐸𝑐𝑒𝑙𝑙 )
2𝐹
(59)
+𝜂𝑎𝑐𝑡,𝑎,𝐻2 𝑗𝑎 𝐴𝑣 + 𝜂𝑎𝑐𝑡,𝑎,𝐶𝑂 𝑗𝑎 𝐴𝑣
Electrolyte
Q=0
(60)
Cathode ERL
𝑄 = 𝜂𝑎𝑐𝑡,𝑐,𝑂2 𝑗𝑐 𝐴𝑣
(61)
Cathode Backing Layer
Q=0
(62)
Cathode Flow Field
Q=0
(63)
Interconnects
Q=0
(64)
6 Chemical Model
6.1 Internal Reforming
When there is methane in the fuel feed of the cell, direct internal reforming occurs where
the methane is converted via the catalyzed methane steam reforming reaction (MSR)
(65), while simultaneously the slightly exothermic water gas shift reaction (WGS) (66)
occurs. Most of the SOFC modeling in the literature today using methane fuels considers
these as the primary two reactions. Several researchers have also investigated the
inclusion the methane carbon dioxide reaction MCDR (67). Another reaction that could
potentially occur in internal reforming SOFCs is the direct steam reforming or
methanation reaction (DSR) (68).
𝐶𝐻4 + 𝐻2 𝑂 ↔ 3𝐻2 + 𝐶𝑂
∆𝐻298 = 206.1 𝑘𝐽/𝑚𝑜𝑙
(65)
𝐶𝑂 + 𝐻2 𝑂 ↔ 𝐻2 + 𝐶𝑂2
∆𝐻298 = −41.2 𝑘𝐽/𝑚𝑜𝑙
(66)
∆𝐻298 = 247 𝑘𝐽/𝑚𝑜𝑙
(67)
𝐶𝐻4 + 𝐶𝑂2 ↔ 2𝐻2 + 2𝐶𝑂
𝐶𝐻4 + 2𝐻2 𝑂 ↔ 4𝐻2 + 𝐶𝑂2
22
∆𝐻298 = 165 𝑘𝐽/𝑚𝑜𝑙
(68)
In this study, the varied fuel compositions consisted of primarily carbon monoxide and
hydrogen, thus only the water gas shift reaction mentioned above was included in the
model. The water gas shift reaction is assumed to take place both in the anode flow
channel and in the anode electrode. The other reactions presented above are discussed in
Appendix A for reference and future studies.
6.2 Chemical Species Balance Equations
Based on consideration of only the water gas shift chemical reaction, the rates of
production and consumption of each species, R rxn,i (kg⁄m3 s) is shown in Table 14
where 𝑟̇𝑟𝑥𝑛 is given in(mol⁄m3 s).
Table 14 Summary of Chemical Species Balance Equations used in Model
Description
Species Balance Equations (kg/m3 ∙ s)
Reaction
𝑅𝐶𝐻4 = 0
−𝑀𝐻2𝑂 𝑟̇𝑊𝐺𝑆
1000
𝑀𝐻2 𝑟̇𝑊𝐺𝑆
𝑅𝐻2 =
1000
−𝑀𝐶𝑂 𝑟̇𝑊𝐺𝑆
𝑅𝐶𝑂 =
1000
𝑀𝐶𝑂2 𝑟̇𝑊𝐺𝑆
𝑅𝐶𝑂2 =
1000
𝑅𝐻2𝑂 =
𝐶𝑂 + 𝐻2 𝑂 ↔ 𝐻2 + 𝐶𝑂2
WGS
6.3 WGS Kinetics
For the water gas shift reaction (WGS) less work has been performed to determine the
optimal kinetic equation. Many authors assume the shift reaction is at equilibrium
however this may not be an accurate assumption. In order to account for a potentially
non-equilibrium state, this study utilizes the equation presented by Haberman and Young
for WGS kinetics [29]. Partial pressures in this equation are in Pascals.
𝑟̇𝑊𝐺𝑆 (𝑚𝑜𝑙 𝑚−3 𝑠 −1 ) = 0.0171 exp (−
𝑝𝐻 𝑝𝐶𝑂2
103191
) (𝑝𝐻2 𝑂 𝑝𝐶𝑂 − 2
)
𝑅𝑇
𝐾𝑒𝑞,2
𝐾𝑒𝑞,2 = 𝑒𝑥𝑝(−0.2935𝑍 3 + 0.6351𝑍 2 + 4.1788𝑍 + 0.3169)
23
(69)
6.4 Carbon Deposition
One of the major challenges in operating internal reforming solid oxide fuel cells is
coking, or carbon formation at the anode inlet. This is detrimental to SOFC performance
as the deposition of carbon particles (coking) on the anode surface can deactivate and
block the catalyst reducing cell performance, impede gas flow and put additional
mechanical stresses on the electrode. The governing reactions for carbon formation in
the fuel cell are via the methane cracking reaction (70) and the Boudouard reaction (71)
with reaction (72) being another probable pathway for carbon formation in the cell.
𝐶𝐻4 ↔ 2𝐻2 + 𝐶
(70)
2𝐶𝑂 ↔ 𝐶𝑂2 + 𝐶
(71)
𝐶𝑂 + 𝐻2 ↔ 𝐻2 𝑂 + 𝐶
(72)
To determine if the cell operating conditions were conducive to carbon formation the
following relationships can be assessed [30].
10614 𝑝𝐶𝐻4
) 2
𝑇
𝑝𝐻2
(73)
2
20634 𝑝𝐶𝑂
𝛼𝑐𝑎𝑟𝑏𝑜𝑛,𝐶𝑂 = 5.744𝑥10−15 𝑒𝑥𝑝 (
)
𝑇
𝑝𝐶𝑂2
(74)
𝛼𝑐𝑎𝑟𝑏𝑜𝑛,𝐶𝐻4 = 4.161𝑥104 𝑒𝑥𝑝 (−
𝛼𝑐𝑎𝑟𝑏𝑜𝑛,𝐶𝑂−𝐻2 = 3.173𝑥10−13 𝑒𝑥𝑝 (
16318 𝑝𝐶𝑂 𝑝𝐻2
)
𝑇
𝑝𝐻2 𝑂
(75)
If the value of 𝛼𝑐𝑎𝑟𝑏𝑜𝑛 is greater than one, the system is not at equilibrium and carbon
can form in the anode. If it is equal to one the system is at thermodynamic equilibrium
and below one carbon formation cannot occur.
24
6.5 Additional Chemical Model Information
Within the model it was necessary to convert from specified inlet mole fractions 𝑥𝑖 to
mass fractions 𝑦𝑖 then species partial pressures 𝑝𝑖 in mixtures. The following equations
were utilized for the conversions assuming ideal gasses.
𝑥𝑖 𝑀𝑖
)
𝑁
∑𝑖=1 𝑥𝑖 𝑀𝑖
(76)
𝑌𝑖
1
( 𝑁
)
𝑀𝑖 ∑𝑖=1 𝑦𝑖 /𝑀𝑖
(77)
𝑦𝑖 = (
𝑥𝑖 =
𝑝𝑖 = 𝑥𝑖 𝑝
(78)
7 Electrochemical Model
Within the fuel cell an electrochemical reaction occurs in which voltage and current are
produced when the anode is supplied with a hydrocarbon and the cathode is supplied
with oxygen, usually in the form of air. The oxygen reacts with the catalyst to produce
oxygen ions which migrate through the electrolyte to the anode. On the anode, hydrogen
or carbon monoxide in the fuel stream reacts with the oxide ions (O2-), producing either
water or carbon dioxide while depositing electrons onto the anode. These electrons pass
through the electrode externally to the fuel cell through the load, and then return to the
cathode.
7.1 Approaches to Electrochemical Modeling
In the literature there are two common approaches utilized in modeling the
electrochemistry of a solid oxide fuel cell. These include 1) the distributed charge
transfer approach and 2) the stepwise subtractive polarization approach. In approach 1,
which is utilized in this study, the charge continuity equation is combined with Ohm’s
law for a balance on the electrochemically active layers of the cell and the potential
gradient across the cell is locally calculated. The overpotential is calculated from the
local potentials in the cell, determined from the charge continuity balances. Both Zhu et
al. and Bessler et al. describe in detail the distributed charge transfer approach [26] [31].
25
In approach 2 the cell voltage is defined as a function of the reversible voltage minus the
combined polarizations (102). Typically, the Butler-Volmer equation is then rearranged
in terms of the activation overpotential such that it is a function of current density to be
substituted into the cell voltage equation, and further equations identified to calculate the
concentration and ohmic overpotential contributions [19].
7.2 Electrochemical Species Balance Equations
The electrochemical reactions occurring in a methane fed solid oxide fuel cell are shown
below. It should be noted that both hydrogen and carbon monoxide oxidation was
included in this study whereas methane oxidation was not, however the methane
oxidation equation is included below for reference.
The following reduction reaction occurs at the cathode ERL of the fuel cell
1
𝑂 + 2𝑒 − → 𝑂2−
2 2
(79)
And the oxidation reactions occurring at the anode ERL of the fuel cell are:
𝐻2 + 𝑂2− → 𝐻2 𝑂 + 2𝑒 −
(80)
𝐶𝑂 + 𝑂2− → 𝐶𝑂2 + 2𝑒 −
(81)
The overall electrochemical reaction in this study is thus:
𝐻2 + 𝐶𝑂 + 𝑂2 → 𝐻2 𝑂 + 𝐶𝑂2
(82)
This paper does not consider the following oxidation reaction of methane in the anode
𝐶𝐻4 + 4𝑂2− → 2𝐻2 𝑂 + 𝐶𝑂2 + 8𝑒 −
(83)
The rates of species consumption due to electrochemical reaction can be represented by
the following equations, with the final equations utilized in this model shown in Table
15.
𝑅𝑒𝑟𝑥𝑛,𝑅𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠 = (−𝜐𝑖 )𝑀𝑖 𝑗𝑖 𝐴𝑣 /𝑛𝑒 𝐹
(84)
𝑅𝑒𝑟𝑥𝑛,𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑠 = (𝜐𝑖 ) 𝑀𝑖 𝑗𝑖 𝐴𝑣 /𝑛𝑒 𝐹
(85)
26
Table 15 Summary of Electrochemical Species Balance Equations used in Model
Species Balance Equations (kg/m3 ∙ s)
Description
Reactions
O2 Red
𝑂2 + 4𝑒 − → 2𝑂2−
H2 Ox
𝐻2 + 𝑂2− → 𝐻2 𝑂 + 2𝑒 −
CO Ox
𝐶𝑂 + 𝑂2− → 𝐶𝑂2 + 2𝑒 −
Overall
𝐻2 + 𝐶𝑂 + 𝑂2 → 𝐻2 𝑂 + 𝐶𝑂2
𝑅𝑂2 =
−𝑀𝑂2 𝑗𝑂2 𝐴𝑣
4𝐹(1000)
𝑅𝐻2𝑂 =
𝑀𝐻2𝑂 𝑗𝐻2 𝐴𝑣
2𝐹(1000)
𝑅𝐻2 = −
𝑀𝐻2 𝑗𝐻2 𝐴𝑣
2𝐹(1000)
𝑅𝐶𝑂 =
−𝑀𝐶𝑂 𝑗𝐶𝑂 𝐴𝑣
2𝐹(1000)
𝑅𝐶𝑂2 =
𝑀𝐶𝑂2 𝑗𝐶𝑂 𝐴𝑣
2𝐹(1000)
Further definition on the electrochemical species balance equations may be found in reference [32]
7.3 Ion and Charge Transfer
For a conducting material, the general equation for the continuity of current and
applicable form of Ohm’s Law combine into the following charge balance equation for
the electron and ion conducting phases in a fuel cell [26] [31]
𝜕𝜌
+ ∇ ∙ 𝒋𝒔 = 𝑅𝑠
𝜕𝑡
where,
𝜕𝜌
𝜕𝑡
(86)
represents the time dependent charge density, 𝒋𝒔 is the current density of the
cell layer where the subscript s = a, c, or e for anode or cathode and electrolyte and 𝑅𝑠 is
the faradaic charge transfer rate (electrical current density source/sink term) for layer s,
where 𝑅𝑠 = 𝑗𝑠 𝐴𝑣 and 𝑗𝑠 is the current density (A/m2). Assuming a steady state case (no
time derivative), and substituting Ohms law into the continuity equation, it can be
rewritten in terms of the effective conductivity (𝜎𝑠𝑒𝑓𝑓 ) and potential (𝜙𝑠 ) for application
towards ionic (io) or electronic (el) potentials in the electrochemically active layers of
the fuel cell.
𝑒𝑓𝑓
∇ ∙ (−𝜎𝑠 ∇𝜙𝑠 ) = 𝑗𝑠 𝐴𝑣
27
(87)
Utilizing the electrochemically active surface area to volume ratio 𝐴𝑣 (m2/m3) is one of
two main methods to account for the conducting particle characteristics in the equation
above, while ensuring the right hand side of the equation reflects a volumetric current
density source or sink term. The other approach is to multiply the exchange current or
current density, which can also be calculated in (A/m) by the triple phase boundary
length 𝑙 𝑇𝑃𝐵 (m/m3). Both of these parameters, 𝐴𝑣 and 𝑙 𝑇𝑃𝐵 , are a function of conducting
particle micro characteristics such as particle radii, volume fraction of particles in
reactive layer, particle coordination numbers, and reactive layer porosity. Both of these
equations are presented in Shi et al. [23]. The conducting particle characteristics are
included in this study via use of the electrochemically active surface are per unit volume
as presented by Kishimoto [20].
7.3.1. Electrode Backing Layers
In the electrode backing layers, there is neither ion transfer, nor electrochemical reaction,
only transfer of electrons via conduction which can be modeled with the following
equations.
𝑒𝑓𝑓
(88)
𝑒𝑓𝑓
(89)
∇ ∙ (−𝜎𝑒𝑙,𝑏𝑎 ∇𝜙𝑒𝑙,𝑎 ) = 0
∇ ∙ (−𝜎𝑒𝑙,𝑏𝑐 ∇𝜙𝑒𝑙,𝑐 ) = 0
𝑒𝑓𝑓
𝑒𝑓𝑓
In these equations, 𝜎𝑒𝑙,𝑏𝑎
and 𝜎𝑒𝑙,𝑏𝑐
are the anode and cathode backing layer effective
electrical conductivities, and 𝜙𝑒𝑙,𝑎 and 𝜙𝑒𝑙,𝑐 are the anode and cathode electrical
potentials. The electronic potentials are the dependent variables assigned to the different
domains, so it is not necessary to identify the electrode with a subscript, as shown in
Table 16.
7.3.2. Electrochemical Reaction Layers (ERL)
One of the assumptions in this model is that the electrochemical reactions occur within a
defined electrochemical reaction layer (ERL) adjacent on either side of the electrolyte.
28
To satisfy this, not only is electron transfer occurring in the ERL but there is a transfer of
ionic species participating in the electrochemical reaction. The mechanism of electron
and ion transfer through the electrodes is modeled by the following equations [15] [33]
[26].
𝑒𝑓𝑓
∇ ∙ (−𝜎𝑒𝑙,𝑎 ∇𝜙𝑒𝑙,𝑎 ) = −𝑗𝑎 𝐴𝑣
(90)
𝑒𝑓𝑓
(91)
𝑒𝑓𝑓
(92)
∇ ∙ (−𝜎𝑖𝑜,𝑎 ∇𝜙𝑖𝑜,𝑎 ) = 𝑗𝑎 𝐴𝑣
∇ ∙ (−𝜎𝑒𝑙,𝑐 ∇𝜙𝑒𝑙,𝑐 ) = 𝑗𝑐 𝐴𝑣
𝑒𝑓𝑓
∇ ∙ (−𝜎𝑖𝑜,𝑐 ∇𝜙𝑖𝑜,𝑐 ) = −𝑗𝑐 𝐴𝑣
(93)
𝑒𝑓𝑓
𝑒𝑓𝑓
In these equations, 𝜎𝑖𝑜,𝑎
and 𝜎𝑖𝑜,𝑐
are the anode and cathode reaction layer effective ionic
conductivities, and 𝜙𝑖𝑜,𝑎 and 𝜙𝑖𝑜,𝑐 are the anode and cathode reaction layer ionic
potentials. The ionic and electronic potentials are the dependent variables assigned to the
different domains, so it is not necessary to identify the electrode with a subscript as
shown in Table 16.
7.3.3. Electrolyte
The electrolyte layer of a SOFC is a dense solid that conducts the oxide ions from the
cathode to the anode. The applicable ion transfer equation for this layer is:
(94)
∇ ∙ (−𝜎𝑖𝑜,𝑒 ∇𝜙𝑖𝑜,𝑒 ) = 0
In this equation, 𝜎𝑖𝑜,𝑒 and 𝜙𝑖𝑜,𝑒 are the electrolyte ionic conductivity and electrolyte ionic
potential. The final form of this equation is shown in Table 16.
Table 16 Summary of Charge Transfer Equations used in Model
𝑒𝑓𝑓
(95)
𝑒𝑓𝑓
(96)
∇ ∙ (−𝜎𝑒𝑙,𝑏𝑎 ∇𝜙𝑒𝑙 ) = 0
Electrode Backing Layers
∇ ∙ (−𝜎𝑒𝑙,𝑏𝑐 ∇𝜙𝑒𝑙 ) = 0
Anode ERL
𝑒𝑓𝑓
∇ ∙ (−𝜎𝑒𝑙,𝑎 ∇𝜙𝑒𝑙 ) = −𝑗𝑎 𝐴𝑣
29
(97)
𝑒𝑓𝑓
(98)
𝑒𝑓𝑓
(99)
∇ ∙ (−𝜎𝑖𝑜 ∇𝜙𝑖𝑜 ) = 𝑗𝑎 𝐴𝑣
Cathode ERL
∇ ∙ (−𝜎𝑒𝑙,𝑐 ∇𝜙𝑒𝑙 ) = 𝑗𝑐 𝐴𝑣
𝑒𝑓𝑓
∇ ∙ (−𝜎𝑖𝑜 ∇𝜙𝑖𝑜 ) = −𝑗𝑐 𝐴𝑣
(100)
∇ ∙ (−𝜎𝑖𝑜 ∇𝜙𝑖𝑜 ) = 0
(101)
Electrolyte
7.4 Cell Voltage
The open circuit or reversible Nernst voltage for any fuel cell is the theoretical
maximum voltage the cell could achieve given a specific set of operating conditions. The
true voltage of the electrochemical cell will not however be equivalent to the open
circuit voltage during operation. The true cell voltage 𝐸𝑐𝑒𝑙𝑙 , is the open circuit voltage
𝐸 𝑟𝑒𝑣 , minus the internal cell resistances and losses or cell polarizations as shown in the
general equation below.
𝑟𝑒𝑣
𝐸𝑐𝑒𝑙𝑙 = 𝐸𝑐𝑒𝑙𝑙
− 𝜂𝑜ℎ𝑚 − 𝜂𝑐𝑜𝑛𝑐 − 𝜂𝑎𝑐𝑡
(102)
As the current is drawn from a cell, the voltage will drop due to the presence of ohmic,
concentration and activation losses. Each of these losses contributes to the total heat
produced in the fuel cell and there are many approaches in the literature on how to
calculate these losses or polarizations. A pictoral representation of the relationship
between ideal open circuit voltage and true cell voltage is shown in Figure 2.
30
Figure 2 Relationship between Ideal and True Cell Voltages [3]
When half cell potentials are considered such as in H2 and CO oxidation the cell voltage
in approach 2 is related by the equation (89) below [18] [14] [19].
𝑟𝑒𝑣
𝐸𝑐𝑒𝑙𝑙 = 𝐸𝑎,𝐻2
− 𝜂𝑎𝑐𝑡,𝐻2 − 𝜂𝑎𝑐𝑡,𝑂2 − 𝜂𝑜ℎ𝑚 − 𝜂𝑐𝑜𝑛𝑐
(103)
𝑟𝑒𝑣
= 𝐸𝑎,𝐶𝑂
− 𝜂𝑎𝑐𝑡,𝐶𝑂 − 𝜂𝑎𝑐𝑡,𝑂2 − 𝜂𝑜ℎ𝑚 − 𝜂𝑐𝑜𝑛𝑐
The cell voltage is satisfied in this model by setting boundary conditions such that 𝜙𝑒𝑙 =
0 in the cathode backing layer (BL) where it interfaces with the cathode flow channel
(FC), and setting 𝜙𝑒𝑙 = 𝐸𝑐𝑒𝑙𝑙 in the anode backing layer (BL) of the cell where it
interfaces with the anode flow channel. For the study 𝐸𝑐𝑒𝑙𝑙 is then varied from -1 to -0.4
[15] [11]. Thus the cell voltage can be written as:
𝐸𝑐𝑒𝑙𝑙 = 𝜙𝑒𝑙,𝑐 |𝐵𝐿−𝐹𝐶 − 𝜙𝑒𝑙,𝑎 |𝐵𝐿−𝐹𝐶
(104)
7.5 Activation Polarizations
The potential of an electrode directly affects the kinetics of the surface reactions. In
electrochemical reactions the activation energy not only includes thermal energy barriers
as in chemical reactions, it must also overcome an electric potential barrier. At low
31
current density and operating temperatures the activation losses may significantly affect
the total voltage of the cell, as the reactants must overcome this activation energy barrier
for the reactions to proceed at the electrodes.
There are two general approaches to determining the relationship between the current
density drawn to the activation overpotential (also referred to as activation polarization).
The first method is more complex and involves consideration of the detailed multistep
elementary reactions on the catalyst for each overall oxidation and reduction equation.
As an example, the overall oxidation of methane on nickel catalyst can be broken down
into 42 separate irreversible reactions to be considered [34]. For simplicity this study
uses the second method.
The second method assumes a single charge transfer reaction or rate limiting reaction
step where the reaction kinetics are described by using the well-known Butler-Volmer
form of the current-overpotential equation, where the general form written in terms of
current density (𝑗 = 𝑖/𝐴) is shown below
𝛼𝑎 𝐹 𝜂𝑎𝑐𝑡
−𝛼𝑐 𝐹 𝜂𝑎𝑐𝑡
𝑗 = 𝑗𝑜 [exp (
) − 𝑒𝑥𝑝 (
)]
𝑅𝑇
𝑅𝑇
(105)
where 𝑗 is current density (A/m2), 𝑗𝑜 is the exchange current density (A/m2), 𝜂𝑎𝑐𝑡 is the
cell activation overpotential (V), and F is Faraday’s constant. In the general ButlerVolmer equation above, α is the transfer coefficient or symmetry factor for each
electrode and when applied to half cell reactions becomes the forward and backward
reaction symmetry factors. The transfer coefficient is used to determine the contribution
of the anode and cathode currents to the total current.
To solve for the cell current density the local activation potentials also need to be
defined. The general form for activation potential in an electrode can be defined as
follows, where 𝐸𝑒𝑞 is the local (s = anode or cathode) electric potential at equilibrium
32
𝜂𝑎𝑐𝑡 = (𝜙𝑒𝑙 − 𝜙𝑖𝑜 ) − 𝐸𝑒𝑞
(106)
Similar SOFC modeling utilizing distributed charge transfer sets the equilibrium
potentials based on an anode reference state set to zero. However in this study, the
reversible Nernst potentials combined with electrode reference potentials utilized by
Suwanwarangkul et al. will be utilized [15].
The Butler-Volmer equation is then applied to each half-cell reaction, where ji is the half
cell local faradaic current density (A/m2), and j0,s,i are the half cell exchange current
densities for the each species based on the exchange current density equations presented
by Suwanwarangkul et al. [15]. For hydrogen oxidation the following equations are used
in this study
2𝐹 𝜂𝑎𝑐𝑡,𝑎,𝐻2
−𝐹 𝜂𝑎𝑐𝑡,𝑎,𝐻2
𝑗𝐻2 = 𝑗𝑜,𝑎,𝐻2 [exp (
) − 𝑒𝑥𝑝 (
)]
𝑅𝑇
𝑅𝑇
11
𝑗𝑜,𝑎,𝐻2 = 2.1 ∙ 10
0.266
𝑅𝑇
𝑝𝐻2𝑂
1.2 ∙ 105
(
)
𝑒𝑥𝑝 (−
)
𝐹 1.78 ∙ 109 𝑝𝐻2
𝑅𝑇
𝑟𝑒𝑣
𝜂𝑎𝑐𝑡,𝑎,𝐻2 = 𝜙𝑒𝑙,𝑎 − 𝜙𝑖𝑜,𝑎 − 𝐸𝑎,𝐻2
𝑟𝑒𝑣
𝐸𝑎,𝐻2
=
𝑝𝐻2 𝑂
𝑅𝑇
ln (
)
2𝐹
1.78 ∙ 109 𝑝𝐻2
(107)
(108)
(109)
(110)
For carbon monoxide oxidation the following equations are used in this study. It should
be noted that the exchange current density for hydrogen oxidation is assumed to be 2.5
times higher than that for carbon monoxide oxidation.
2𝐹 𝜂𝑎𝑐𝑡,𝑎,𝐶𝑂
−𝐹 𝜂𝑎𝑐𝑡,𝑎,𝐶𝑂
𝑗𝐶𝑂 = 𝑗𝑜,𝑎,𝐶𝑂 [exp (
) − 𝑒𝑥𝑝 (
)]
𝑅𝑇
𝑅𝑇
𝑗𝑜,𝑎,𝐶𝑂 = 0.84 ∙ 1011
0.266
𝑝𝐶𝑂2
𝑅𝑇
1.2 ∙ 105
(
)
𝑒𝑥𝑝
(−
)
𝐹 1.63 ∙ 109 𝑝𝐶𝑂
𝑅𝑇
(111)
(112)
𝑟𝑒𝑣
𝜂𝑎𝑐𝑡,𝑎,𝐶𝑂 = 𝜙𝑒𝑙,𝑎 − 𝜙𝑖𝑜,𝑎 − 𝐸𝑎,𝐶𝑂
(113)
𝑝𝐶𝑂2
𝑅𝑇
ln (
)
2𝐹
1.63 ∙ 109 𝑝𝐶𝑂
(114)
𝑟𝑒𝑣
𝐸𝑎,𝐶𝑂
=
33
For oxygen reduction the following equations are used in this study
𝑗𝑂2 = 𝑗𝑜,𝑐,𝑂2 [exp (
−2𝐹 𝜂𝑎𝑐𝑡,𝑐,𝑂2
2𝐹 𝜂𝑎𝑐𝑡,𝑐,𝑂2
) − 𝑒𝑥𝑝 (
)]
𝑅𝑇
𝑅𝑇
11
𝑗𝑜,𝑐,𝑂2 = 0.25 ∙ 10
𝑅𝑇
1.3 ∙ 105
0.5
(𝑝 ) 𝑒𝑥𝑝 (−
)
𝐹 𝑂2
𝑅𝑇
𝑟𝑒𝑣
𝜂𝑎𝑐𝑡,𝑐,𝑂2 = 𝜙𝑒𝑙,𝑐 − 𝜙𝑖𝑜,𝑐 − 𝐸𝑐,𝑂2
𝑟𝑒𝑣
𝐸𝑐,𝑂2
=
𝑅𝑇
ln(𝑝𝑂2 )
4𝐹
(115)
(116)
(117)
(118)
The current density of the cell can be evaluated as either the ion current through the
electrolyte or by the charge transfer rates in either the anode or cathode yielding the
following relations.
𝑗𝑐 = 𝑗𝑂2
𝑗𝑎 = 𝑗𝐻2 + 𝑗𝐶𝑂
(119)
7.6 Ohmic Polarizations
The ohmic losses in a fuel cell are due to the ionic resistance in the electrolyte combined
with the resistance for the electrons passing through the electrodes and current
collectors. Ohmic losses through an electrolyte can be reduced by decreasing the
thickness of the electrolyte or increasing its ionic conductivity. The ohmic loss, which is
the potential difference across the electrolyte, is included in this study via the charge
continuity equations as the effective conductivity term presented previously for each
layer. To calculate the effective conductivities the following Arrhenius form equations
are utilized [12]
𝜎𝑒𝑙,𝑎 =
𝜎𝑒𝑙,𝑐 =
95𝑥106
𝑇
42𝑥106
𝑇
exp (−
exp (−
1150
) for Ni-YSZ
(120)
) for LSM-YSZ
(121)
10300
(122)
𝑇
1200
𝑇
𝜎𝑖𝑜 = 3.34 𝑥104 exp (−
34
𝑇
) for YSZ
The temperature based calculated electronic and ionic conductivities above are utilized
to calculate the effective conductivity values for use in the charge transfer model. The
symbol 𝑉𝑒𝑙 𝑜𝑟 𝑖𝑜 is the volume fraction of electron or ion conducting particles [10].
Table 17 Summary of Effective Conductivity Equations used in Model
1 − 𝜀𝑎
𝑒𝑓𝑓
Electrode Backing Layers
𝜎𝑒𝑙,𝑏𝑎 = 𝜎𝑒𝑙,𝑎 (
)
𝜏𝑒𝑙,𝑎
(123)
1 − 𝜀𝑐
)
𝜏𝑒𝑙,𝑐
(124)
𝑒𝑓𝑓
𝜎𝑒𝑙,𝑏𝑐 = 𝜎𝑒𝑙,𝑐 (
Anode ERL
1 − 𝜀𝑎
𝑒𝑓𝑓
𝜎𝑒𝑙,𝑎 = 𝜎𝑒𝑙,𝑎 (
) 𝑉𝑒𝑙,𝑎
𝜏𝑒𝑙,𝑎
𝑒𝑓𝑓
1 − 𝜀𝑎
𝜎𝑖𝑜,𝑎 = 𝜎𝑖𝑜 (
) 𝑉𝑖𝑜,𝑎
𝜏𝑖𝑜,𝑎
Cathode ERL
1 − 𝜀𝑐
𝑒𝑓𝑓
(125)
(126)
𝜎𝑒𝑙,𝑐 = 𝜎𝑒𝑙,𝑐 (
) 𝑉𝑒𝑙,𝑐
𝜏𝑒𝑙,𝑐
(127)
1 − 𝜀𝑐
) 𝑉𝑖𝑜,𝑐
𝜏𝑖𝑜,𝑐
(128)
𝑒𝑓𝑓
𝜎𝑖𝑜,𝑐 = 𝜎𝑖𝑜 (
𝜎𝑖𝑜
Electrolyte
(129)
7.7 Concentration Polarizations
In addition to the activation losses, the concentration losses at each electrode must be
considered. Concentration losses are due to the physical variation in species from the
flow channels to the ERL, where the electrochemical reaction occurs. This location is
assumed to be at the interface between the electrolyte and the electrode. They can be due
to diffusion of species through the cell surfaces from the bulk flow path or the transport
of species through the electrodes. The losses in the cathode are typically small when
compared to losses at the anode. In this model, the concentrations at the boundary
between the electrodes and electrolyte (ERL) are handled by creating separate
electrochemical reaction layers. Within these layers, the partial pressures are calculated
for the present species using the Maxwell Stefan approach outlined in section 4 Mass
Transfer. Based on this approach, including the concentration overpotential term in the
calculation of the cell potential is not required [14] [19].
35
8 Results
8.1 Solution Method
All of the equations defined in the modeling sections previously presented are
simultaneously solved using COMSOL Multiphysics FEM modeling software. The mesh
consisted of 34,400 elements in total, with a higher density of elements at the inlet and
outlet of the cell, in the ERL’s and in the electrolyte layer as shown in Figure 3. To
achieve the varying distribution along the cell, the length was divided into 400 elements
with an arithmetic sequence, symmetric distribution (in reverse direction) using an
element ratio of 0.01. The remaining distributions for each subdomain in the vertical
direction utilized a fixed number of evenly distributed elements.
Figure 3 Distribution of Model Mesh Elements for Initial 1/10th of Total Cell Length
To solve the system, a segregated step PARDISO solver was used with an overall
relative tolerance of 0.001. A consistent stabilization tuning parameter of Ck=0.4 was
implemented in the model along with an automatic highly nonlinear damping method
terminating based on tolerance for each set of variables solved for. The order of the
segregated steps included; 1) velocity and pressure 2) electronic/ionic potentials 3)
cathode species distribution 4) anode species distribution and 5) temperature. To
generate the polarization curves, parametric voltage steps were applied in increments of
0.05 from -0.95V to -0.6V.
36
The length of time for the solution of each case varied from 1 to 3 hours. In general, the
duration of time to achieve convergence increased as the values of the local current
densities increased. Thus, cases with higher amounts of H2 and CO participating in the
electrochemical reaction, which generated higher current densities, took longer to
converge (especially at the lower applied voltages).
In addition, significantly high current densities and high reaction rates introduced
instability into the model. This is true for the electrochemical reaction rates, chemical
reaction rates and heat generation rates in the system. In the case where very large
current densities were calculated (such as near the ERL-electrolyte interface), then
introduced into the electrochemical reaction and electrochemical heat source term
equations, this resulted in unreasonably high rates of change that the software had
difficulty managing and the software would not converge on a solution. Similarly, for
the chemical reactions, when the reaction rate became very large, the system results
oscillated out of control resulting in non-convergence.
Given the presence of these instabilities, in order to attain convergence in all cases run in
the model, a dampening constant of 0.05% was applied to the electrochemical species
generation and electrochemical heat generation source term equations. This dampening
constant serves to promote convergence as well as adjust the polarization curve to align
with experimental data as shown in the next section. In addition, most cases were limited
to a lowest applied voltage of 0.60V due to high current densities or high WGS reaction
rates.
8.2 Comparison to Previous Studies
In order to ensure the results of the model compared well with existing experimental
data, Figure 4 shows the results from Suwanwarangkul et al. for various simulated
syngas fuels at 800oC . Case F5 depicts syngas with a composition of 0.32 H2, 0.03 H2O,
0.45 CO, 0.15 CO2, and 0.03 N2 and most closely aligns with cases 1 and 4 in this study.
37
The data from case F5 in this plot was extracted and compared against both cases 1 and
4 from this study with the results shown in Figure 5.
Figure 4 Polarization Curves with Model (lines) and Experimental data (symbols) in Literature [15]
Figure 5 Comparison Between Case 1 and 4 with Data from Literature
38
Additionally, the dampening factor of 0.05% was varied to determine the impact on the
polarization curve with the results shown in Figure 6. Note that the compositions
between Case 1 and the experimental data varied slightly.
Figure 6 Effect of Varying Dampening Factor on Case 1 Polarization Curve
8.3 Velocity and Pressure Profiles
The anode and cathode inlet velocity and outlet pressure boundary conditions were the
same for all cases modeled. Shown below is the inlet velocity and pressure profiles for
𝐸𝑐𝑒𝑙𝑙 = 0.7. In all cases, there was no notable change in the velocity or pressure profile
distributions and very small velocity magnitude differences at each of the different cell
voltages applied and cases run as shown in Table 18. The inlet effects shown in the
example case 1 velocity distribution in Figure 7 occur at a distance between 0 m and
2x10-4m (0.2% of total cell length) and are due to a non-parabolic inlet velocity profile
boundary condition applied at the inlet as shown in Figure 8.
39
Figure 7 Case 1 Inlet Velocity Profile (m/s) for Ecell=0.7
Figure 8 Case 1 Inlet Velocity Boundary Condition Profiles (m/s) for Ecell=0.7
The systems were operated at pressures slightly above ambient with an outlet pressure
boundary condition of 1 atm. The inlet effects shown at the inlet to both the anode and
cathode electrode layers in Figure 9 is due to species mass diffusion limitations at the
modeled permeability. The smaller atoms permeate or diffuse much faster into the
40
porous media than the larger atoms and this results in a localized change in pressure.
These inlet effects, like those occurring in the velocity distribution, occur in the initial
0.2% of total cell length.
Figure 9 Case 1 Inlet Pressure Distribution (Pa) for Ecell=0.7
Table 18 Comparison of Velocity and Pressure Maximums for each Case at Ecell=0.7
Case
1
2
3
4
5
Max Velocity
(m/s)
10.237
10.21
10.215
10.225
10.207
Max Pressure
(kPa)
291.01
290.99
291.0
291.0
290.98
8.4 Species Distribution
In the model there were changes in species composition along the length of the cell due
to the water gas shift chemical reaction as well as the electrochemical oxidations of
hydrogen and carbon monoxide in the anode and reduction of oxygen in the cathode.
Figure 10 through Figure 29 shows the mole fractions of the primary and reacting
species in the anode inlet (first 15% of total length) for an applied 𝐸𝑐𝑒𝑙𝑙 = 0.7.
41
When comparing the individual species distribution results from the different cases the
water gas shift reaction is performing as expected and the reacting species undergo
concentration changes in the anode flowfield. For hydrogen, the least change in mole
fraction across the cell and most similar distributions occur in cases 1 and 4, with case 5
displaying a similar distribution but a more notable difference in the bulk electrode and
flowfield mole fractions. In cases 2 and 3 the mole fraction close to the inlet in the
electrode is notably greater before decreasing along the ERL side of the electrode. As
expected in all cases, there is a decrease in H2 mole fraction from electrochemical
oxidation towards the anode ERL.
For carbon monoxide the largest gradients occurring in the flowfield due to the water gas
shift reaction are in case 2 and 3. At the very inlet to the electrode there is an increased
CO mole fraction due to mass diffusion limitations before the system moves towards
equilibrium as the species consumption due to electrochemical oxidation in the ERL
dominates in the electrode.
Of all the species present in the fuel, carbon dioxide has the largest molecular mass.
Based on the theory that the utilized permeability value in the porous media limits the
larger molecules from penetrating as quickly into the electrodes, the observed inlet
effects would dictate that there would be an initial decrease in CO2 mole fraction in
order to balance the initial increases in mole fraction from other species. This is indeed
the case as best shown in Figure 20. Otherwise, cases 1 and 4 display very similar
species distributions for CO2. In cases 2 and 3 the greatest increase in species mole
fraction in the flowfield is observed. As expected, the CO2 mole fraction increases
towards the anode ERL due to CO electrochemical oxidation.
With regards to water, the greatest decrease in water mole fraction in the flowfield due
to the water gas shift reaction was observed in cases 2 and 3. Inlet effects in the
electrode resulted in a lower initial mole fraction before the system moves towards
equilibrium and H2 oxidation results in an increased mole fraction of water towards the
42
anode ERL. Interestingly it was observed that the water mole fraction in the flowfield for
case 5 slightly increased, which is unlike all the other cases run.
Figure 10 Case 1 Inlet H2 Mole Fractions for Ecell=0.7
Figure 11 Case 1 Inlet CO Mole Fractions for Ecell=0.7
Figure 12 Case 1 Inlet CO2 Mole Fractions for Ecell=0.7
43
Figure 13 Case 1 Inlet H2O Mole Fractions for Ecell=0.7
Figure 14 Case 2 Inlet H2 Mole Fractions for Ecell=0.7
Figure 15 Case 2 Inlet CO Mole Fractions for Ecell=0.7
Figure 16 Case 2 Inlet CO2 Mole Fractions for Ecell=0.7
44
Figure 17 Case 2 Inlet H2O Mole Fractions for Ecell=0.7
Figure 18 Case 3 Inlet H2 Mole Fractions for Ecell=0.7
Figure 19 Case 3 Inlet CO Mole Fractions for Ecell=0.7
Figure 20 Case 3 Inlet CO2 Mole Fractions for Ecell=0.7
45
Figure 21 Case 3 Inlet H2O Mole Fractions for Ecell=0.7
Figure 22 Case 4 Inlet H2 Mole Fractions for Ecell=0.7
Figure 23 Case 4 Inlet CO Mole Fractions for Ecell=0.7
Figure 24 Case 4 Inlet CO2 Mole Fractions for Ecell=0.7
46
Figure 25 Case 4 Inlet H2O Mole Fractions for Ecell=0.7
Figure 26 Case 5 Inlet H2 Mole Fractions for Ecell=0.7
Figure 27 Case 5 Inlet CO Mole Fractions for Ecell=0.7
Figure 28 Case 5 Inlet CO2 Mole Fractions for Ecell=0.7
47
Figure 29 Case 5 Inlet H2O Mole Fractions for Ecell=0.7
The water gas shift reaction is assumed to occur both in the anode flowfield and the
anode electrode layers. The rates of the water gas shift reaction throughout the anode are
shown below. There was a very small change in the WGS reaction rate at different
applied voltages for the same case. However there was a significant increase in the rate
of the reaction as the water content was increased in case 3 and as expected, case 2
shows the second highest WGS reaction rate due to inlet water concentration. Cases 4
and 5 show the lowest reaction rates, with the increase in CO2 concentration driving the
reaction closer to equilibrium.
Figure 30 Case 1 Rate of WGS Reaction (mol/m3-s) for Ecell=0.7
48
Figure 31 Case 2 Rate of WGS Reaction (mol/m3-s) for Ecell=0.7
Figure 32 Case 3 Rate of WGS Reaction (mol/m3-s) for Ecell=0.7
Figure 33 Case 4 Rate of WGS Reaction (mol/m3-s) for Ecell=0.7
Figure 34 Case 5 Rate of WGS Reaction (mol/m3-s) for Ecell=0.7
49
At each applied voltage, the probability of carbon formation was assessed. Due to the
low concentration of methane in the fuel inlet, the probability of carbon formation due to
the methane cracking reaction was neglected in this study. It was observed for all cases,
the carbon activity did not exceed a value of one, such that carbon formation in this cell
for the conditions examined, remained thermodynamically impossible. This is likely due
to the lack of extreme temperature gradients as would be seen in a cell fed with methane,
undergoing the methane steam reforming reaction.
For the calculated carbon activities in this study, the highest carbon activities are
observed at the anode electrode inlet. As the applied cell voltage increases (and current
density decreases), the probability of carbon formation throughout the electrode
increases slightly. The highest carbon activities occurred in case 1, at the highest applied
cell voltage of 0.95V. For the Boudouard reaction, the largest observed carbon activity
was approximately 0.925, much greater than the carbon activities found in the other
cases. For cases 1,4 and 5 the carbon activity decreased moving in the y-direction from
the anode flowfield to the ERL. Cases 2 and 3 showed the least amount of change in
carbon activity throughout the electrode.
Figure 35 Case 1 Carbon Formation Activity in the Anode via the Boudouard Reaction for E cell=0.7
50
Figure 36 Case 1 Carbon Formation Activity in the Anode via the Boudouard Reaction for Ecell=0.95
Figure 37 Case 2 Carbon Formation Activity in the Anode via the Boudouard Reaction for Ecell=0.95
Figure 38 Case 3 Carbon Formation Activity in the Anode via the Boudouard Reaction for Ecell=0.95
Figure 39 Case 4 Carbon Formation Activity in the Anode via the Boudouard Reaction for Ecell=0.95
51
Figure 40 Case 5 Carbon Formation Activity in the Anode via the Boudouard Reaction for Ecell=0.95
For reaction (72), similar to the Boudouard reaction, the highest carbon activities
occurred in case 1, at the largest applied cell voltage of 0.95V. The calculated carbon
activity was less than that found in the Boudouard reaction, with the largest observed
value being 0.766 for case 1 with larger carbon activities found at the electrode inlet.
The least amount of change in the y-direction across the electrode occurred in case 5 and
case 2.
Figure 41 Case 1 Carbon Formation Activity in the Anode via reaction (72) for Ecell=0.7
Figure 42 Case 1 Carbon Formation Activity in the Anode via reaction (72) for Ecell=0.95
52
Figure 43 Case 2 Carbon Formation Activity in the Anode via reaction (72) for Ecell=0.95
Figure 44 Case 3 Carbon Formation Activity in the Anode via reaction (72) for Ecell=0.95
Figure 45 Case 4 Carbon Formation Activity in the Anode via reaction (72) for Ecell=0.95
Figure 46 Case 5 Carbon Formation Activity in the Anode via reaction (72) for Ecell=0.95
53
8.5 Temperature
The typical temperature distribution for all cases considered is shown below. The
temperature increases in the cell are due to the contributions from the water gas shift and
electrochemical reactions. The maximum temperatures attained varied only slightly from
case to case as shown in Table 19. It was found that the highest temperature increases
over the length of the cell were in case 1 when the lowest voltage was applied (0.6V).
This is likely due to the largest concentrations of H2 and CO at the inlet resulting in the
highest current density of all the cases at similar voltages. This would result in an
increase in the rate of electrochemical heat generation in the cell.
Figure 47 Case 1 Temperature Distribution for Ecell=0.7
Figure 48 Case 1 Temperature Distribution for Ecell=0.6
Table 19 Comparison of Maximum Temperatures for each Case at Ecell=0.7
Case
1
2
3
4
5
Max
Temperature
(K)
1036.1
1033.5
1034
1035
1033.3
54
8.6 Electrochemistry
The cell polarization curves, shown below, were determined by integrating the total
volumetric current density across the anode ERL for each applied voltage. Included in
the plot for the individual polarization curves is the calculated reversible or open circuit
voltages for both the H2 and CO fuels. For modeling, the applied cell voltages were
selected such that they do not exceed the calculated reversible potentials as is expected
in normal operation.
As expected, due to the larger concentrations of H2 and CO in the fuel stream, case 1
shows the greatest performance while due to the lowest concentrations, case 3 shows the
worst performance. The polarization curves for case 2 and case 5 demonstrate the same
performance while case 4 which contains less water but a greater amount of N2 than case
2, demonstrates the second highest performance.
Figure 49 Polarization Curves for Case 1 to Case 5
55
Figure 50 Case 1 Polarization Curve
Figure 51 Case 2 Polarization Curve
56
Figure 52 Case 3 Polarization Curve
Figure 53 Case 4 Polarization Curve
57
Figure 54 Case 5 Polarization Curve
The local current density values in the cell were calculated based on contributions from
both hydrogen and carbon monoxide electrochemical oxidation. Within the anode ERL,
the current density is primarily zero until approximately 0.001mm adjacent to the
electrolyte layer where it sharply increases as shown below. As noted previously, the
anode ERL is 0.03mm in height, ranging from 1.58mm at the electrolyte interface to
1.61mm at the electrode interface. Therefore, most of the current generated in the cell
due to the electrochemical oxidation of H2 and CO occurs within the initial 1.7% to 3.3%
of the total ERL length from the electrode-electrolyte interface.
Additionally, the local current density at the ERL inlet is lower than that at the ERL
outlet. Neglecting the inlet effects present in this study the inlet current density would
expected to be lower than the values currently shown when compared to the outlet
current densities.
58
Figure 55 Case 1 Total Local Current density across Anode ERL in y-direction for Ecell=0.7
Figure 56 Case 2 Total Local Current density across Anode ERL in y-direction for Ecell=0.7
59
Figure 57 Case 3 Total Local Current density across Anode ERL in y-direction for Ecell=0.7
Figure 58 Case 4 Total Local Current density across Anode ERL in y-direction for Ecell=0.7
60
Figure 59 Case 5 Total Local Current density across Anode ERL in y-direction for Ecell=0.7
The calculated local current density across the cell in the x-direction at the anode ERLelectrolyte interface is shown below. The initial sharp change in the current density
occurs in the first 0.002m of the cell length (0.2% of total length) and is a function of the
pressure and species inlet effects presented previously.
Figure 60 Case 1 Local Current Densities at Anode ERL-Electrolyte Interface for Ecell=0.7
61
Figure 61 Case 2 Local Current Densities at Anode ERL-Electrolyte Interface for Ecell=0.7
Figure 62 Case 3 Local Current Densities at Anode ERL-Electrolyte Interface for Ecell=0.7
62
Figure 63 Case 4 Local Current Densities at Anode ERL-Electrolyte Interface for Ecell=0.7
Figure 64 Case 5 Local Current Densities at Anode ERL-Electrolyte Interface for Ecell=0.7
9 Conclusion
This study examined the effects of varying simulated syngas inlet feed conditions on a 2D multi-physics model of a solid oxide fuel cell operating with an inlet temperature of
63
800oC. When comparing the resulting polarization curves to experimental data it was
found that the current model aligns well with the experimental data taken from
Suwararungkul et al. [15]. Further optimization of the dampening parameter for an even
closer fit to the experimental data could be performed however for the purposes of this
case comparison study it is not required.
When moving along the polarization curve for the electrochemical cell, several
mechanisms were shown to occur. Generally it was shown that as the applied cell
voltage is decreased and the cell current density increases; the heat generation within the
cell increases, and the probability for carbon formation decreases.
Observable inlet effects were shown in the model, most notably the mass diffusion
limitations present when a permeability value of 2.5x10-14m2 was utilized for the porous
electrodes. These effects only occurred in the initial 0.2% of total cell length.
When comparing the varying feed compositions, it was found that the inlet fuel
concentrations of CO and H2 had the most significant effect on cell performance. This
resulted in case 1 having the best performance and case 3 the lowest performance.
Increasing the water content in the fuel decreased the probability for carbon formation
although in this study carbon formation would not be expected to occur due to lack of
large temperature gradients and the cell not being operated below 250 A/m2. Case 4,
which contained a larger balance of nitrogen actually performed better than cases 2, 3
and 5. Cases 2 and 5 in which 10% of the feed was changed from water to carbon
dioxide showed very similar performances.
10 Future Work
Multiphysics modeling of solid oxide fuel cells at the cell level presents many challenges
due to the complex nature of interacting momentum, mass, and heat transfer combined
with complex electrochemistry. In most cases, improving the accuracy of the model will
directly result in an increase in computational demand. Therefore depending on the
64
required accuracy of the model, assumptions must be carefully balanced with the
available hardware for computations as well as necessary speed to run model cases.
In future studies, the model presented in this paper may be modified to include a
parabolic laminar inlet flow profile, optimized dampening parameter, elementary
reaction
mechanisms
for
both
the
chemical
and
electrochemical
reactions,
electrochemical kinetic equations that have been validated against multiple sets of
experimental data, inclusion of additional reforming reactions, the use of alternative
fuels and modifying the anode with materials to reduce carbon formation.
In order to scale the model up to a stack and system level in future studies, it would be
recommended to perform a sensitivity study of various aspects of the model and make
assumptions to reduce the level of detail in order to reduce computational demand.
65
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70
12 Appendix A
MSR Kinetics
In this study, the methane steam reforming reaction was investigated for potential
inclusion in the model. Due to computational constraints it was not included in the final
model however the research performed on the reaction kinetics has been included here
for reference and future studies.
According to Mogensen et al. [35] and Nagel et al. [36], in modeling methane steam
reforming kinetics, there exists a wide variation in the equations formulated. This is
likely due to the different operating conditions of the experiments as well as conditions
not being allowed to reach steady state before data is taken. Nagel [36] notes that most
of the discrepancies in the kinetic equations are in the reaction orders for water. It is
noted that S/C ratios << 2 yield positive reaction orders, S/C ratios ~ 2 yield a zero
reaction order and larger S/C ratios > 2 yield a negative reaction order in the MSR steam
reforming kinetic equations.
There are also studies which utilize the elementary steps of the reactions to predict the
reaction kinetics [34]. This approach can become highly computationally intensive
especially when modeling multiple types of physics, therefore it is recommended in
complex modeling cases to focus on using a singular rate equation for each of the
reforming reactions noted above.
There are three types of kinetic rate expressions currently used for the MSR including;
General Langmuir-Hinshelwood kinetics, First order reaction in methane, and Power law
expressions derived from data fitting [35]. In order to determine the appropriate kinetic
models for this study, the available models in the literature were reviewed.
In general Langmuir-Hinshelwood kinetic models (Type 1) for the reforming reaction
focus on the rate determining steps in species surface reactions and the generation of
kinetic equations is a direct result of which mechanistic reaction steps are assumed or
71
determined to be rate determining steps. In 1989 Xu and Froment proposed a steam
reforming reaction rate that was based on the rate determining step of the reaction of
adsorbed carbon and oxygen species utilizing the partial pressures of methane, water and
hydrogen [37] and their work has been utilized in many studies [14]. Similar to Xu and
Froment, Lehnert [38] and Haberman [29] also include a first order dependence on
water. According to Mogensen et. al. [35], the presence and thus the effect of the partial
pressure value for water in the numerator of these equations is not commonly observed
in experiments. Newer studies have been performed that identify the rate limiting step as
the dissociative adsorption of methane with a reaction order of 1 which is a generally
agreed upon approach. However there is disagreement on whether other rate limiting
steps in the mechanism should contribute to the kinetic models. Mogensen et al. also
notes that these kinetic equations are subject to the operating conditions as well since
different reaction steps become rate controlling depending on operating temperatures.
Thus, in order to utilize the Langmuir-Hinshelwood kinetic models, it is suggested to use
a model that was developed with similar operating conditions as the experiment or
model under development.
First order kinetic models (Type 2) are Langmuir-Hinshelwood kinetic models
considering only the methane dissociative adsorption as the rate determining step [6]
[39]. One of the most commonly utilized MSR kinetic equations used in Ni-YSZ SOFC
studies is that proposed by Achenbach [39]. Although considering only methane
simplifies the equation and eliminates the concern for inconsistencies due to additional
rate limiting steps needing consideration, Mogensen suggests these rate equations are
only valid at high temperatures and low pressures.
In the last set and most mathematically simple kinetic models commonly proposed for
the reforming reaction, a power law equation (Type 3) is fit to individual experimental
conditions by measuring catalytic reaction rates [40]. The general form for the power
law equation is [35]:
72
𝛽
𝛾
𝛼
𝛿
𝜀
−𝑟̇𝐶𝐻4 = 𝑘 𝑝𝐶𝐻4
𝑝𝐻2𝑂 𝑝𝐻2 𝑝𝐶𝑂2
𝑝𝐶𝑂
𝑒𝑥𝑝 (−
𝐸𝑎
)
𝑅𝑇
(130)
In Appendix B, some commonly used rate equations for the Ni-YSZ methane steam
reforming (MSR) reaction are listed.
MCDR Kinetics
MCDR kinetic rate equations with regards to Ni-YSZ materials are difficult to find in
the literature and there are few fuel cell modeling studies considering the MCDR
reaction. In their model, Ni utilizes a Languir-Hinshelwood type equation that was taken
from experimental data of CO2 reforming of methane on Ru/Al2O3 catalyzed metallic
foam absorber [19]. As an alternative equation for modeling MCDR kinetics, Verykios
developed a kinetic equation based on dry reforming of methane over Ni/La2O3 catalyst
[41].
DSR Kinetics
It is well known that the methanation reaction (DSR) will only occur at temperatures
below 675oC and thus this reaction is not included in most modeling efforts. This
assumption may not be correct however with the large temperature gradients in the cell.
When the DSR or methanation reaction is included, the most popular kinetics rate
equations are of the Languir-Hinshelwood type by Xu and Froment [37] [14] and Hou
and Hughes [42].
73
Table 20 Kinetic Models for SOFC MSR and WGS Reactions on Ni Catalysts
Equations
+
−
3
𝑟̇𝑀𝑆𝑅 = 𝑘𝑀𝑆𝑅
𝑝𝐶𝐻4 𝑝𝐻2𝑂 − 𝑘𝑀𝑆𝑅
𝑝𝐶𝑂 𝑝𝐻2
T
(oC)
750 900
P
(bar)
1.5
750 900
1.5
800 900
1
300 575
3 - 15
S/C
1
+
−
𝑟̇𝑊𝐺𝑆 = 𝑘𝑊𝐺𝑆
𝑝𝐶𝑂 𝑝𝐻2𝑂 − 𝑘𝑊𝐺𝑆
𝑝𝐶𝑂2 𝑝𝐻2
𝑟̇𝑀𝑆𝑅 = 𝑘1 (𝑝𝐶𝐻4 𝑝𝐻2𝑂 −
𝑝𝐶𝑂 𝑝𝐻2 3
𝐾𝑒𝑞,1
)
𝑘1 = 2395 exp(−231266/𝑅𝑇)
3
𝐾𝑒𝑞,1 = 1.0267 ∗ 1010 𝑒𝑥𝑝(−0.2513𝑍 4 + 0.3665𝑍 3 + 0.5810𝑍 2 − 27.134𝑍 + 3.2770)
𝑟̇𝑊𝐺𝑆 = 𝑘2 (𝑝𝐻2𝑂 𝑝𝐶𝑂 −
𝑝𝐻2 𝑝𝐶𝑂2
) 𝑚𝑜𝑙 𝑚−3 𝑠 −1
𝐾𝑒𝑞,2
Material
Type
50%wt ZrO2, 50%wt Ni
2 mm thick cermet (CH4
reforming zone 0.15 to
0.3mm)
50%wt ZrO2, 50%wt Ni
2 mm thick cermet (CH4
reforming zone 0.15 to
0.3mm)
Note: Experimental Data
used from Lehnert et al.
Type 1
30%wt ZrO2, 70%wt Ni
10 µm thick anode on 2mm
thick YSZ disk
Tubular reactor of 15.2%
Ni and MgAl2O4 catalyst
Type 1
[38]
Type 1
[29]
𝑘2 = 0.0171 exp(−103191/𝑅𝑇) 𝑚𝑜𝑙 𝑚−3 𝑃𝑎 −2 𝑠 −1
𝐾𝑒𝑞,2 = 𝑒𝑥𝑝(−0.2935𝑍 3 + 0.6351𝑍 2 + 4.1788𝑍 + 0.3169)
𝑟̇𝑀𝑆𝑅 = 𝑘𝑎𝑑 𝑝𝐶𝐻4 (1 −
𝑟̇𝑀𝑆𝑅 = 𝑘1 (
𝑘𝑎𝑑 𝑝𝐻2 𝑝𝐶𝐻4
)
𝑘𝑟 𝐾𝐻2𝑂 𝑝𝐻2𝑂
𝑝𝐶𝐻4 𝑝𝐻2𝑂 𝑝𝐶𝑂 𝑝𝐻2 0.5
−
) /𝐷𝐸𝑁 2
𝑝𝐻2 2.5
𝐾𝑒𝑞,1
17
𝐾𝑒𝑞,1 = 1.198 ∗ 10
𝑟̇𝑊𝐺𝑆 = 𝑘2 (
exp(−26830/𝑇)
𝑝𝐶𝑂 𝑝𝐻2𝑂 𝑝𝐶𝑂2
−
) /𝐷𝐸𝑁 2
𝑝𝐻2
𝐾𝑒𝑞,2
𝐾𝑒𝑞,2 = 1.767 ∗ 10−2 exp(4400/𝑇)
𝑟̇𝐷𝑆𝑅 = 𝑘3 (
𝑝𝐶𝐻4 𝑝𝐻2 2𝑂 𝑝𝐶𝑂2 𝑝𝐻2 0.5
−
) /𝐷𝐸𝑁 2
𝑝𝐻2 3.5
𝐾𝑒𝑞,3
𝐾𝑒𝑞,3 = 2.117 ∗ 1015 exp(−22430/𝑇)
𝐷𝐸𝑁 = 1 + 𝐾𝐶𝑂 𝑝𝐶𝑂 + 𝐾𝐻2 𝑝𝐻2 + 𝐾𝐶𝐻4 𝑝𝐶𝐻4 + 𝐾𝐻2𝑂 𝑝𝐻2𝑂 /𝑝𝐻2
74
0-2
3, 5
[43]
Type 1
[37]
𝑟̇𝑀𝑆𝑅 = 𝑘1 (
𝑝𝐶𝐻4 𝑝𝐻2𝑂 0.5
𝑝𝐶𝑂 𝑝𝐻2 3
)
(1
−
) /𝐷𝐸𝑁 2
𝑝𝐻2 1.5
𝑝𝐶𝐻4 𝑝𝐻2𝑂 𝐾𝑒𝑞,1
17
𝐾𝑒𝑞,1 = 1.198 ∗ 10
𝑟̇𝑊𝐺𝑆 = 𝑘2 (
598 823
1.2 6
4-7
Tubular reactor of 8385%wt Al2O, 15-17%wt Ni
Type 1
700 940
1.1 2.8
2.6 8
80%wt ZrO2, 20%wt Ni
1.4mm thick cermet
Type 2
650 950
-
2
35%wt ZrO2, 65%wt Ni
40 µm thick anode
Type 2
854 907
1
Ni- ZrO2 50 µm thick anode
Type 3
[42]
exp(−26830/𝑇)
𝑝𝐶𝑂 𝑝𝐻2𝑂 0.5
𝑝𝐶𝑂2 𝑝𝐻2
) (1 −
) /𝐷𝐸𝑁 2
0.5
𝑝𝐻2
𝑝𝐶𝑂 𝑝𝐻2𝑂 𝐾𝑒𝑞,2
𝐾𝑒𝑞,2 = 1.767 ∗ 10−2 exp(4400/𝑇)
𝑟̇𝐷𝑆𝑅 = 𝑘3 (
𝑝𝐶𝐻4 𝑝𝐻2𝑂
𝑝𝐶𝑂2 𝑝𝐻2 4
)
(1
−
) /𝐷𝐸𝑁 2
𝑝𝐻2 1.75
𝑝𝐶𝐻4 𝑝𝐻2𝑂 2 𝐾𝑒𝑞,3
𝐾𝑒𝑞,3 = 2.117 ∗ 1015 exp(−22430/𝑇)
𝐷𝐸𝑁 = 1 + 𝐾𝐶𝑂 𝑝𝐶𝑂 + 𝐾𝐻2 𝑝𝐻2 0.5 + 𝐾𝐻2𝑂 𝑝𝐻2 𝑂 /𝑝𝐻2
𝑝𝐶𝑂 𝑝𝐻3 2
𝐸𝐴
) 𝑒𝑥𝑝 (− )
𝑝𝐶𝐻4 𝑝𝐻2𝑂 𝐾𝑒𝑞
𝑅𝑇
𝑟̇𝑀𝑆𝑅 = 𝑘1 𝑝𝐶𝐻4 (1 −
𝐸𝐴 = 82000
𝐽
[39]
𝑘1 = 4274𝑚𝑜𝑙 𝑠 −1 𝑚 −2 𝑏𝑎𝑟 −1
𝑚𝑜𝑙
𝑟̇𝑀𝑆𝑅 = 𝑘1 𝑝𝐶𝐻4 (1 −
3
𝑝𝐶𝑂 𝑝𝐻
2
𝑝𝐶𝐻4 𝑝𝐻2 𝑂 𝐾𝑒𝑞
) 𝑒𝑥𝑝 (−
𝐸𝐴
𝑅𝑇
) 𝐸𝐴 = 63300
𝐽
𝑚𝑜𝑙
𝑘1 =
[6]
.00498 𝑚𝑜𝑙 −1 𝑠 −1 𝑚−2 𝑃𝑎 −1
𝛽
𝛼
−𝑟𝑀𝑆𝑅 = 𝑘 𝑝𝐶𝐻
𝑝
𝑒𝑥𝑝 (−
4 𝐻2 𝑂
𝐸𝑎 = 95 ± 2
𝑘𝐽
𝑚𝑜𝑙
𝐸𝑎
𝑅𝑇
)
𝛼 = 0.85 ± 0.05
𝛽 = −0.35 ± 0.04
𝑘 = 8542 𝑚𝑜𝑙 𝑠 −1 𝑚−2 𝑏𝑎𝑟 −1
75
1.53 2.5
[40]
13 Appendix B
Table 21 Sensitivity Analysis of Calculated Diffusion Coefficients
Temperature
(K)
1073.15
873.15
1273.15
1073.15
1073.15
1073.15
1073.15
1073.15
Porosity
0.5
0.5
0.5
0.3
0.3
0.5
0.5
0.5
Tortuosity
2
2
2
2
5
5
2
2
Pore
Diameter
(µm)
1
1
1
1
1
1
1
2
Pressure
(Pa)
101325
101325
101325
101325
101325
101325
202650
101325
CH4-CO
2.041E-4
1.423E-4
2.752E-4
2.041E-4
2.041E-4
2.041E-4
1.020E-4
2.041E-4
CH4-H2O
2.466E-4
1.719E-4
3.326E-4
2.466E-4
2.466E-4
2.466E-4
1.233E-4
2.466E-4
CH4-H2
6.626E-4
4.619E-4
8.936E-4
6.626E-4
6.626E-4
6.626E-4
3.313E-4
6.626E-4
CH4-CO2
1.672E-4
1.165E-4
2.255E-4
1.672E-4
1.672E-4
1.672E-4
8.360E-5
1.672E-4
CO-H2O
2.447E-4
1.706E-4
3.300E-4
2.447E-4
2.447E-4
2.447E-4
1.224E-4
2.447E-4
CO-H2
7.395E-4
5.154E-4
9.972E-4
7.395E-4
7.395E-4
7.395E-4
3.697E-4
7.395E-4
CO-CO2
1.542E-4
1.075E-4
2.080E-4
1.542E-4
1.542E-4
1.542E-4
7.712E-5
1.542E-4
H2O-H2
8.509E-4
5.931E-4
1.147E-3
8.509E-4
8.509E-4
8.509E-4
4.254E-4
8.509E-4
H2O- CO2
1.965E-4
1.370E-4
2.650E-4
1.965E-4
1.965E-4
1.965E-4
9.825E-5
1.965E-4
H2-CO2
6.230E-4
4.343E-4
8.402E-4
6.230E-4
6.230E-4
6.230E-4
3.115E-4
6.230E-4
O2 -N2
1.936E-4
1.350E-4
2.611E-4
1.936E-4
1.936E-4
1.936E-4
9.680E-5
1.936E-4
CH4-CO
8.462E-9
7.632E-9
9.217E-9
5.077E-9
2.031E-9
3.385E-9
8.460E-9
1.692E-8
CH4-H2O
9.624E-9
8.680E-9
1.048E-8
5.774E-9
2.310E-9
3.850E-9
9.622E-9
1.924E-8
CH4-H2
1.322E-8
1.192E-8
1.440E-8
7.931E-9
3.172E-9
5.287E-9
1.322E-8
2.643E-8
CH4-CO2
7.247E-9
6.537E-9
7.894E-9
4.348E-9
1.739E-9
2.899E-9
7.246E-9
1.449E-8
CO-H2O
8.279E-9
7.467E-9
9.018E-9
4.967E-9
1.987E-9
3.312E-9
8.278E-9
1.656E-8
CO-H2
1.025E-8
9.246E-9
1.117E-8
6.150E-9
2.460E-9
4.100E-9
1.025E-8
2.050E-8
CO-CO2
6.618E-9
5.969E-9
7.209E-9
3.971E-9
1.588E-9
2.647E-9
6.617E-9
1.323E-8
H2O-H2
1.255E-8
1.132E-8
1.367E-8
7.530E-9
3.012E-9
5.020E-9
1.255E-8
2.510E-8
H2O- CO2
7.131E-9
6.432E-9
7.768E-9
4.279E-9
1.712E-9
2.853E09
7.130E-9
1.426E-8
H2-CO2
8.279E-9
7.468E-9
9.018E-9
4.968E-9
1.987E-9
3.312E09
8.279E-9
1.656E-8
𝑀𝑆
𝐷𝑖𝑗
𝑒𝑓𝑓
𝐷𝑖𝑗
76
7.250E-9
6.539E-9
7.897E-9
4.350E-9
1.740E-9
2.900E09
7.249E-09
1.450E-8
CH4-CO
3.385E-8
3.054E-8
3.687E-8
3.385E-8
3.385E-8
3.385E-8
3.385E-8
6.771E-8
CH4-H2O
3.850E-8
3.473E-8
4.194E-8
3.850E-8
3.850E-8
3.850E-8
3.850E-8
7.700E-8
CH4-H2
5.287E-8
4.769E-8
5.759E-8
5.287E-8
5.287E-8
5.287E-8
5.287E-8
1.057E-7
CH4-CO2
2.899E-8
2.615E-8
3.158E-8
2.899E-8
2.899E-8
2.899E-8
2.899E-8
5.799E-8
CO-H2O
3.312E-8
2.987E-8
3.607E-8
3.312E-8
3.312E-8
3.312E-8
3.312E-8
6.624E-8
CO-H2
4.100E-8
3.699E-8
4.466E-8
4.100E-8
4.100E-8
4.100E-8
4.100E-8
8.201E-8
CO-CO2
2.648E-8
2.388E-8
2.884E-8
2.648E-8
2.648E-8
2.648E-8
2.648E-8
5.295E-8
H2O-H2
5.020E-8
4.528E-8
5.468E-8
5.020E-8
5.020E-8
5.020E-8
5.020E-8
1.004E-7
H2O- CO2
2.853E-8
2.573E-8
3.108E-8
2.853E-8
2.853E-8
2.853E-8
2.853E-8
5.706E-8
H2-CO2
3.312E-8
2.987E-8
3.607E-8
3.312E-8
3.312E-8
3.312E-8
3.312E-8
6.624E-8
O2 -N2
2.900E-8
2.616E-8
3.159E-8
2.900E-8
2.900E-8
2.900E-8
2.900E-8
5.801E-8
O2 -N2
𝑘𝑛
𝐷𝑖𝑗
77
14 Appendix C
Table 22 Polarization Curve Data Table Case 1 to 5 Dampening 0.07% vs Literature Values
Applied
Voltage
Case 1
Case 2
Case 3
Case 4
Case 5
Suwan.
Exp. Data
[15]
Suwan. Model
Data
[15]
1.0
-
-
-
-
-
160
20
0.95
315.87
280.82
260.32
305.65
276.40
285
90
0.9
421.88
377.87
352.89
406.05
372.94
385
150
0.85
542.23
487.71
459.66
521.00
485.39
490
220
0.8
678.47
609.21
581.92
648.13
615.79
590
315
0.75
830.29
741.18
716.08
788.43
762.32
700
425
0.7
997.12
880.26
861.54
936.48
927.71
805
545
0.65
1169.47
1026.98
-
1090.58
1109.42
-
675
0.6
-
-
-
1248.56
-
-
-
Table 23 Polarization Curve Data Table Case 1 to 5 Dampening 0.05% vs Literature Values
Applied
Voltage
Case 1
Case 2
Case 3
Case 4
Case 5
Suwan.
Exp. Data
[15]
Suwan. Model
Data
[15]
1.0
-
-
-
-
-
160
20
0.95
306.88
275.70
253.22
299.24
272.55
285
90
0.9
403.56
365.83
341.12
393.22
362.05
385
150
0.85
511.29
467.23
440.92
497.81
462.84
490
220
0.8
628.81
578.12
551.20
611.71
573.81
590
315
0.75
752.54
697.10
670.70
731.61
693.73
700
425
0.7
883.81
822.83
798.22
858.69
821.34
805
545
0.65
1020.71
953.99
932.58
990.95
955.39
-
675
0.6
1159.26
1087.27
1068.81
1124.84
1091.80
-
-
78